3 GEOMETRY AND MEASUREMENT CLAST MATHEMATICS COMPETENCIES IB1: IB2a: IB2b: IB2c: IIB1: IIB2: IIB3: IIB4: IIIBl: IIIB2: IVB1: IVB2: Round measurements to the nearest given unit of the measuring device used Calculate distances Calculate areas Calculate volumes Identify relationships between angle measures Classify simple plane figures by recognizing their properties Recognize similar triangles and their properties Identify appropriate units of measurement for geometric objects Infer formulas for measuring geometric figures Identify applicable formulas for measuring geometric figures Solve real-world problems involving perimeters, areas, volumes of geometric figures Solve real-world problems involving the Pythagorean theorem 132 3.1 MEASURING DISTANCE, AREA AND VOLUME In this Chapter we shall discuss the measurement of distances, areas, volumes and their applications. We shall then learn about the measurement of angles, the relationship between different types of angles, the classification of plane figures and similar triangles. A. Rounding Measurements Objective IB1 1. 2. 3. 4. CLAST SAMPLE PROBLEMS Round 43.846 to the nearest hundredth of a centimeter Round 19,283 tons to the nearest ten thousand tons Round 9 pounds, 6 ounces to the nearest tenth of a pound Round the measurement of the screw to the nearest 1/4 inch In Section 1.3 we learned how to round numbers. We are now ready to round measurements in such units as feet, yards, centimeters, meters, pounds, and liters. 1 ROUNDING MEASUREMENTS RULE EXAMPLES 1. Underline the place to which you are rounding. 2. If the first number to the right of the underlined place is 5 or more, add one to the underlined number. Otherwise, do not change the underlined number. 3. Change all the numbers to the right of the underlined number to zeros. Note: We are asking to round 3138 pounds to the nearest hundred pounds. The idea and procedure would be the same if you were asked to round 3138 grams to the nearest hundred grams or 3138 liters to the nearest hundred liters. ANSWERS 1. 43.85 cm Round 38.67 centimeters to the nearest centimeter. Since we want to round to the nearest centimeter, we underline the units place. 1. Underline the 8. 38.67 2. The first number to the right of 8 is 6, so we add one to 8. (8 + 1 = 9) 3. Change all the numbers to the right of 8 to zeros obtaining 39.00 or 39. Round 3138 pounds to the nearest hundred pounds. 1. Underline the hundreds place. 3138 2. The first number to the right of 1 is 3. Do not change the underlined number. 3. Change all the numbers to the right of 1 to 0's obtaining 3100. Thus, 3138 pounds rounded to the nearest hundred pounds becomes 3100 pounds. 2. 20,000 tons Copyright © Houghton Mifflin Company. All rights reserved. 3. 9.4 pounds 4. 3/4 inch SECTION 3.1 Measuring Distance, Area and Volume 133 CLAST EXAMPLE Example 1. Round the measurement of the length of the 1 paper clip to the nearest 4 inch. A. 1 in. 1 B. 14 in. 3 C. 14 in. D. 2 in. Solution Note: The abbreviation for inch is in. 1 If we are measuring (or counting) by 4 in., the 1 2 3 measures would be: 4 in., 4 in., 4 in., 4 5 4 in., 4 in., and so on. The end of the paper 5 clip is closest to the 4 in. mark. 5 1 Since 4 in. = 14 in., the correct answer is B. B. Calculating Distances, Areas and Volumes Objectives IB2a,IB2b,IB2c CLAST SAMPLE PROBLEMS 1. 3. 5. 7. 8. 9. Change 4 yards to feet 2. Change 8 1/2 ft to yards Convert 9.7 kilometers to meters 4. Convert 90 centimeters to hectometers Change 18 inches to yards 6. Change 2.2 miles to feet Find the perimeter of a rectangle that has a length of 4 yards and a width of 4 feet Find the distance around the polygon shown in kilometers 300 m 600 m Find the distance around a regular pentagon with one side having a 400 m length of 6 meters 500 m 10. What is the distance around a circular walk with a diameter of 6 ft. 11. What is the area in square centimeters of the triangle shown? 30 cm 12. What is the area of a circular region with a 6 yard diameter? 40 mm 13. Find the area of the 3 meter wide rectangle shown 13 cm 14. What is the surface area of a rectangular solid that is 6 feet long, 1 yard wide and 4 feet high? 15. Find the volume of a right circular cone 3 meters high and with a 5m radius of 20 centimeters 3m 16. What is the volume of a rectangular solid 5 feet long, 4 feet wide and 9 inches high? 17. Find the volume of a sphere with a diameter of 2 yards 18. What is the volume in liters of a 850-milliliter bottle? 19. What is the volume in cubic centimeters of a 25.8-liter container? 20. What is the volume in cubic meters of a 675 -liter container? ANSWERS 1. 12 ft 5. 1/2 yd 9. 30 m 13. 12 m2 4 17. 3 π yd3 2. 2 5/6 yd 6. 11,616 ft 10. 6 π ft 14. 108 ft2 3. 9700 meters 7. 32 ft 11. 60 cm2 15. 40,000 π cm3 4. 0.009 hm 8. 1.8 km 12. 9p yd2 16. 15 ft3 18. 0.85 L 19. 25,800 cm3 20. 0.675 m3 Copyright © Houghton Mifflin Company. All rights reserved. 134 CHAPTER 3 Geometry and Measurement When we measure a distance we measure length and the result is a linear unit. In the U. S. system we use inches, feet, yards and miles. In the metric system we use millimeters, centimeters, meters, and kilometers. Here are some relationships you should know. T TERMINOLOGY -- LINEAR MEASURES U. S. UNITS EXAMPLES 12 inches (in.) = 1 foot (ft) 52 1 1 52 inches = 52 ft = 12 ft = 4 3 ft. 1 12 1 (in.) = 12 ft 3 ft = 1 yard (yd) 1 1 ft = 3 yd 8 ft 7920 ft 5280 ft = 1 mile (mi) 1 1 ft = 5280 mi METRIC UNITS kilometer hectometer dekameter meter (km ) (hm) (dam) m 1000 m 100m 10 m meter decimeter m (dm) 1 10 m centimeter millimeter (cm) (mm) 1 1 100 m 1000 m Chart: km hm dam m dm cm mm = 8 8 2 1 yd = 3 yd = 2 3 yd. 3 = 7920 7920 1 mi = 5280 mi 5280 1 = 1 2 mi. To convert from one metric unit to another is a matter of moving the decimal point. For example, to convert from km to meters, we move three places right in the chart. Thus, to convert 73 km to meters, we move the decimal point in 73. three places right obtaining 73. km = 73,000 m. To convert 3847 mm to meters, we have to move the decimal point in 3847. three places left. Thus, 3847. mm = 3.847 m. You can recall the order in the metric chart if you remember that: King Henry Died Monday Drinking Chocolate Milk! 2 FINDING PERIMETERS RULE A polygon is a closed plane figure having three or more sides. (See figure at right.) EXAMPLES 9 in 10 in 5 in. 15 in The perimeter of (distance around) a polygon is the sum of the lengths of its sides. The perimeter of the polygon, in inches, is (10 + 9 + 5 + 15) in. = 39 in. The perimeter 3 1 39 in feet is 12 = 3 12 = 3 4 ft. Copyright © Houghton Mifflin Company. All rights reserved. SECTION 3.1 Measuring Distance, Area and Volume 3 135 FINDING CIRCUMFERENCES The distance around a circle is called the circumference C and given by the formula C = π d = 2π r radius r diameter d Note that the diameter d of the circle is twice d the radius, that is, d = 2r or r = 2 . Note: The CLAST uses π as a factor in its answers, you need not approximate π If the radius of the circle is 7 ft, (r = 7) the circumference C = 2 π r = 2 π (7) = 14 π ft. CLAST EXAMPLE Solution Example 2. What is the distance around the polygon, in meters? 75 cm 78 cm The distance around the polygon is: 95 + 75 + 80 + 78 = 328 cm Since 100 cm is 1 m, 328 cm = 3.28 m. (If you look in the chart, to go from cm to m we have to go two places left, so we move the decimal point in 328. two places left obtaining 3.28 m.) The answer is C. 80 cm 95 cm A. 328 m B. 32.8 m C. 3.28 m D. 0.328 m Hint: When adding the length of the sides, we added 95 + 75 by writing 75 as 5 + 70 and then adding 95 + 5 + 70 = 170. Then add 80 to 170, obtaining 250 and finally, add 78 to get 328. Be careful with the arithmetic! Sometimes we have to use more than one formula to get the perimeter (distance around) of a figure. For example, the next problem involves a rectangle and two semicircles (half circles). In order to find the distance around the figure we need to know the formulas for the perimeter of a rectangle and the perimeter (circumference) of a circle. Copyright © Houghton Mifflin Company. All rights reserved. 136 CHAPTER 3 Geometry and Measurement CLAST EXAMPLES Example Solution 3. The figure below shows a running track in the shape of a rectangle with semi-circles at each end. The total distance around the track is: Straight Distance + Distance around the two semicircles The straight distance is 3x + 3x or 6x and for convenience, you can think of the two semicircles as a complete circle of diameter d = 2y as shown. 2y 3x What is the distance around the track? 2y A. 6x + 4y + 2 π y B. 2 π y + 6x C. π D. π y2 + 3x The circumference C of this circle is C = π d = π •2y = 2 π y y2 + 6x Note that the straight distance around the track is 3x + 3x = 6x. Thus, the answer must involve 6x. Eliminate C. Thus, the total distance around the track is: Straight Distance + 6x + Distance around the two semicircles 2π y Since 6x + 2 π y = 2 π y + 6x, the answer is B. To find the area of a plane figure such as a circle, square, rectangle, parallelogram or triangle, we need the formulas that follow. Note: When we measure area or surface area the result is in square units (abbreviated as sq. units or units2.) 4 FINDING AREAS RULE EXAMPLES 2 The area of a circle of radius 5 in. is: A=π r The area of a circle of radius r is: 2 A = π •5 sq. in. = 25 π sq. in. The area of a square of side s is: A = s2 The area of a square of side 3 m is: A = 32 sq. m = 9 sq. m The area of a rectangle of base b and height h is: A = bh The area of a rectangle of base 4 mm and height 8 mm is: A = 4•8 sq. mm = 32 sq. mm The area of a triangle of base b and height h is: 1 A = 2 bh The area of a triangle of base 2 yd and height 4 yd is 1 A = 2 •2•4 = 4 sq. yd Copyright © Houghton Mifflin Company. All rights reserved. SECTION 3.1 Measuring Distance, Area and Volume 137 CLAST EXAMPLE Example Solution 4. What is the area of a circular region whose diameter is 6 centimeters? A. 36 π sq. cm B. 6 π sq. cm C. 12 π sq. cm D. 9 π sq. cm The formula to find the area of a circle is A = π r2. Since the diameter is 6 cm, the radius is half of that, or 3 cm. Thus, the area is: A = π •(3)2 sq. cm = 9 π sq. cm. The answer is D. (Some books use cm2 but the CLAST uses sq. cm.) In case the answer to the problem requires conversions (cm to meters or in. to ft), do the conversions before you use the correct area formula, as shown in the next example. CLAST EXAMPLE Example Solution 5. What is the area of a triangle with a base of 24 inches and a height of 12 inches? A. 12 sq. ft B. 1 sq. ft C. 24 sq. ft D. 144 sq. ft Note that the answers are given in square feet. 5 First, convert the original dimensions to feet. 24 in. = 2 ft, 12 in. = 1 ft. The area of the 1 triangle is A = 2 bh, where b = 2 and h = 1. 1 Thus, A = 2 •2•1 = 1 sq. ft. The answer is B. FINDING SURFACE AREAS The surface area of a rectangular solid of length L, width W, and height H is: A = 2HW + 2LH + 2WL Note that to obtain the surface area of the solid, we first need the area of the front and the back (2HW), the two sides (2LH) and the top and bottom (2WL). The surface area of a rectangular solid 4 ft long, 3 ft wide and 2 ft high is: H L W A = 2•(2•3) + 2•(4•2) + 2•(3•4) = 12 + 16 + 24 = 52 sq. ft Copyright © Houghton Mifflin Company. All rights reserved. 138 CHAPTER 3 Geometry and Measurement CLAST EXAMPLE Example Solution 6. What is the surface area of a rectangular solid that is 12 inches long, 5 inches wide and 6 inches high? Since we are looking for a surface area the answer must be in square units (eliminate choices A and C). A. B. C. D. The area of the front is The area of the back is The area of the right side is The area of the left side is The area of the top is The area of the bottom is The total surface area is: 360 cubic inches 324 square inches 324 cubic inches 360 square inches Note: Draw a picture whenever possible! 6 12 6•5 = 30 sq. in. 6•5 = 30 sq. in. 12•6 = 72 sq. in. 12•6 = 72 sq. in. 5•12 = 60 sq. in. 5•12 = 60 sq. in. 324 sq. in. The answer is B. 5 When we measure volume we measure the amount of space in a solid three dimensional object and the result is in cubic units, sometimes written as units3. In the U.S. system we have cubic inches, cubic feet and cubic yards. The metric system uses cubic centimeters, cubic meters and so on. The volume of liquids in the metric system is measured in liters. T TERMINOLOGY -- METRIC CUBIC MEASURES To convert from one metric unit to another METRIC UNITS is a matter of moving the decimal point. For kiloliter hectoliter dekaliter liter example, to convert from kL to liters, we move (kL) (hL) (daL) L three places right in the chart. Thus, to convert 1000 L 100L 10 L 37 kL to liters, we move the decimal point in liter deciliter centiliter milliliter 37. three places right obtaining L (dL) (cL) (mL) 37. kL = 37,000 L. 1 1 1 To convert 2647 mL to liters, we have to move 10 L 100 L 1000 L the decimal point in 2647. three places left. Thus, 2647. mL = 2.647 L. Chart: kL hL daL L dL cL mL King Hendy Died Late Drinking Chocolate Milk CLAST EXAMPLE Example 7. Find the volume in centiliters (cl) of a 2.95 liter bottle. A. 29.5 cl C. 0.295 cl B. 295 cl D. 2950 cl Solution Since centi means 100, one liter = 100 cl. Thus, 2.95 liter s= 2.95•100 cl = 295 cl. The answer is B. You do not need any formulas but remember your prefixes like milli, centi, deci and kilo. Copyright © Houghton Mifflin Company. All rights reserved. SECTION 3.1 Measuring Distance, Area and Volume 139 To find the volume of a rectangular solid (a box), right circular cylinder, circular cone or sphere we need the formulas that follow. 6 H FINDING VOLUMES RULE EXAMPLES The volume of a rectangular solid The volume of a rectangular solid of length L, width W and height H is: 10 ft long, 9 ft wide and 8 ft high is: V = 10•9•8 cubic ft = 720 cubic ft V = LWH L W r The volume of a right circular cylinder of radius r and height h is: h V = π r 2h V = π r 2 h = π •52•7 cubic cm = 175 π cubic cm The volume of a right circular cone of radius r and height h is: h 1 V = π r 2h 3 r The volume of a sphere of radius r is: 4 V = π r3 3 r The volume of a right circular cylinder of radius 5 cm and height 7 cm is: The volume of a right circular cone of radius 6 in and height 12 in is: 1 1 V = 3 π r2h = 3 π •62•12 cubic in. = 144 π cubic in. The volume of a sphere of radius 3 m is 4 4 V = 3 π r3 = 3 π •33 cubic m 4 = 3 π •27 cubic m = 36 π cubic m CLAST EXAMPLE Example Solution 8. What is the volume of a sphere with a 12 in. diameter? Since the volume must be in cubic in., eliminate B and D. Since the diameter is 12 in, the radius is 6 in. 4 Substituting 6 for r in V = 3 π r3 we obtain: 4 V = 3 π •216 cubic in = 288 π cubic in. The answer is C. A. 48 π cubic in. B. 48 π sq. in. C. 288 π cubic in. D. 288 π sq. in. Caution: To use the formula for the volume of a sphere you need the radius r. Copyright © Houghton Mifflin Company. All rights reserved. 140 CHAPTER 3 Geometry and Measurement C. Identifying the Appropriate Unit of Measurement Objective IIB4 CLAST SAMPLE PROBLEMS 1. Which unit of measurement would be used to give the amount of wall surface to be covered by the contents of a can of paint? A. gallons B. liters C. square feet D. cubic feet 2. Which unit of measurement would be used to find the distance around a building? A. liters B. cubic feet C. square meters D. meters 3. What unit of measurement would be used to find the amount of paper needed for the label on the can? A. inches B. square inches C. cubic inches D. fluid ounces 7 RULE PROPER MEASUREMENT UNITS EXAMPLES When measuring length use linear units (kilometers, miles, meters, yards, centimeters, inches, or millimeters) The diameter of a coin may be measured in inches or centimeters. The distance around a circular track may be measured in meters or yards. When measuring area use square units. (Square yards, square meters, square feet, square centimeters) The amount of paper needed to wrap a gift may be in square inches. The amount of wall surface covered by a gallon of paint may be in square meters. When measuring volumes use cubic units. (Cubic centimeters, cubic inches, cubic feet, cubic meters, gallons, liters) The amount of gasoline in a fuel storage tank may be measured in cubic feet, cubic meters, gallons or liters. CLAST EXAMPLE Example 9. Which of the following would not be used to measure the amount of water needed to fill a swimming pool? A. Cubic feet C. Gallons ANSWERS B. Liters D. Meters 1. C Solution We are to measure the volume of the water. Since volume is measured in cubic units, meters is not an appropriate unit of measure. The answer is D. 2. D Copyright © Houghton Mifflin Company. All rights reserved. 3. B SECTION 3.1 Measuring Distance, Area and Volume 141 Section 3.1 Exercises WARM-UPS A 1. Round the measurement of the length of the key to the nearest inch. 2. 1 Round the measurement of the length of the key to the nearest 2 inch. CLAST PRACTICE A PRACTICE PROBLEMS: Chapter 3, # 1-2 1 3. Round the measurement of the length of the key to the nearest 4 inch. A. 2 in. B. 1 24 in. C. 1 2 2 in. D. 3 24 in. WARM-UPS B 4. 5. 6. Find the distance around the polygon in meters. 150 m 200 m 250 m Find the distance around the polygon in kilometers. 300 m 20 ft Find the distance around the polygon in feet. 40 ft 7. 30 ft Find the distance around the polygon in yards. 35 ft 8. Find the circumference of a circle of radius 8 cm. 9 Find the circumference of a circle whose diameter is 10 in. 10. Find the area of a circle of radius 10 m. Copyright © Houghton Mifflin Company. All rights reserved. 142 CHAPTER 3 Geometry and Measurement 11. Find the area of a circle whose diameter is 12 ft. 12. Find the area of a rectangle of base 8 mm and height 10 mm. 13. Find the area of a rectangle of base 2 yards and height 4 yards. 14. Find the area of a triangle of base is 3 km and height 4 km. 15. Find the area of a triangle of base 2 ft and height 7 ft. 16. Find the surface area of a rectangular solid of length 8 cm, width 10 cm and height 7 cm. 17. Find the surface area of a shoe box that is 12 inches long, 8 inches wide and 4 inches high. 18. Find the volume of the shoe box in Problem 17. 19. Find the volume of a rectangular solid of length 30 mm, width 20 mm and height 10 mm. 20. Find the volume of a right circular cylinder of radius 8 in and height 9 in. 21. Find the volume of a right circular cylinder whose diameter is 10 cm and whose height is 12 cm. 22. Find the volume in cubic feet of a right circular cylinder of radius 6 feet and height 30 inches. 23. Find the volume in cubic feet of a sphere whose diameter is 24 inches. 24. Find the volume in cubic yards of a sphere whose radius is 6 feet. 25. Find the volume in centiliters (cl) of a 1.95 liter bottle. 26. Find the volume in centiliters (cl) of a 3.90 liter bottle. Copyright © Houghton Mifflin Company. All rights reserved. SECTION 3.1 Measuring Distance, Area and Volume CLAST PRACTICE B 27. 143 PRACTICE PROBLEMS: Chapter 3, # 3-8 Find the distance around the polygon in kilometers A. 2300 km B. 23 km C. 2.3 km D. 230 km 600 m 300 m 400 m 400 m 600 m 28. Find the circumference of a circle of radius 12 in. A. 24 π sq. in. B. 144 π in. C. 24 π in. D. 22 π in. 29. What is the area of a circular region of diameter 10 centimeters? A. 20 π sq. cm B. 25 π sq. cm, C, 10 π sq. cm. D. 100 π sq. cm. 30. What is the area of a circular region of radius 6 ft? A. 12 π sq. ft B. 24 π sq. ft. C. 36 π sq. ft D. 144 π sq. ft. 31. What is the area of a square that has sides 6 centimeters in length? A. 24 sq. cm B. 12 sq. cm C. 25 sq. cm. D. 36 sq. cm 32. Find the area of a rectangle that is 6 inches long and 4 inches wide. A. 20 sq. in. B. 24 sq. in. C. 10 sq. in. D. 30 sq. in. 33. Find the area of a triangle with base 8 in. and height 10 in. A. 40 sq. in. B. 80 sq. in. C. 40 in. D. 80 in. 34. Find the surface area of a rectangular solid that is 14 inches long, 5 inches wide and 4 inches high. A. 292 sq. in. B. 650 in. C. 900 sq. in. D. 650 sq. in. Copyright © Houghton Mifflin Company. All rights reserved. 144 CHAPTER 3 Geometry and Measurement 35. Find the volume of a rectangular solid that is 7 cm long, 5 cm wide and 4 cm high. A. 140 sq. cm B. 168 cubic cm C. 140 cubic cm D. 166 cm 36. Find the volume of a right circular cylinder 7 inches high and with a 5 in. radius. A. 35 π cubic in. B. 42 π cubic in. C. 175 π sq. in. D. 175 π cubic in. 37. Find the volume of a right circular cone of radius 9 ft and height 10 ft. A. 270 π sq. ft B. 810 π cubic ft C. 270 π cubic ft D. 870 π sq. ft 38. Find the volume of a sphere with a radius of 9 mm. A. 108 π sq. mm B. 972 π cubic mm C. 18 π mm D. 729 π cubic mm WARM-UPS C 39. What type of measure is needed to express the circumference of a circle? 40. What type of measure is needed to express the diameter of a coin? 41. What type of measure is needed to express the area of a rectangle? 42. What type of measure is needed to express the area of a triangle? 43. What type of measure is needed to express the amount of water in a pool? 44. What type of measure is needed to express the amount of gas in your gas tank? CLAST PRACTICE C PRACTICE PROBLEMS: Chapter 3, # 9-11 45. What type of measure is needed to express the length of the line segment AC of the given rectangle? A. Square B. Linear C. Cubic D. Equilateral 46. What type of measure is needed to express the volume of the rectangular solid? A. Square B. Linear C. Cubic D. Equilateral Copyright © Houghton Mifflin Company. All rights reserved. A B C D H L W SECTION 3.1 Measuring Distance, Area and Volume 145 47. What type of measure is needed to express the surface area of the rectangular solid? A. Square B. Linear C. Cubic D. Equilateral H L W 48. What type of measure is needed to express the thickness of a coin? A. Square B. Linear C. Cubic D. Equilateral 49. What type of measure is needed to express the amount of paint in a can? A. Square B. Linear C. Cubic D. Equilateral 50. What type of measure is needed to express the amount of sand in a box? A. Square B. Linear C. Cubic D. Equilateral EXTRA CLAST PRACTICE 51. What is the area of the figure in square meters? A. 1400 square meters B. 14 square meters C. 14 square meters D. 24 square meters 140 cm 10m 52. What is the area of a triangle whose base is 36 feet and whose height is 40 inches? A. 60 square feet B. 180 square feet C. 360 square feet D. 720 square feet Copyright © Houghton Mifflin Company. All rights reserved. 146 CHAPTER 3 Geometry and Measurement 3.2 APPLICATIONS AND PROBLEM SOLVING The information we have learned in Section 3.1 will now be used to solve problems involving perimeters, areas, and volumes. In addition, we shall learn how to select and create new formulas to measure other geometric figures. If you have forgotten about the RSTUV procedure for solving problems, review it before you go on. A. Applications and the Pythagorean Theorem Objectives IVB1, IVB2 CLAST SAMPLE PROBLEMS 1. A new street connecting A Street and B Avenue is to be constructed. Construction costs are about $100 per linear foot. What is the estimated cost for constructing the new street? B Avenue New 12 miles A Street 5 miles 2. A football player runs 8 yards to the right as shown. He then turns left, runs down field and catches a 10 yard pass thrown from the spot from where he started. How far down field did he run? 10 yd d=? 8 yd CLAST EXAMPLES Example Solution 1. What will be the cost of tiling a room measuring 12 feet by 15 feet if square tiles costing $2 each and measuring 12 inches on each side are used? A. $180 B. $4320 C. $360 D. $3600 Note: The measurements of the tile and of the room are not in the same units. It is easier to change the measurement of the tile to feet. Since 12 inches = 1 ft, the tile measures 1 ft on each side. If the tiles were 8 in. on each side, you would need to convert 8 in. to ft. ANSWERS 1. $6,864,000 1. Read the problem. You are given the dimensions of the room and the cost of each tile. 2. Select the unknown. We want to find the cost of tiling the room. 3. Think of a plan. Find how many tiles are needed and multiply by the cost of each tile ($2). 4. Use the fact that we need to cover a rectangle measuring 12 tiles by 15 tiles. Thus, we need 12 × 15 = 180 tiles. Total cost = Cost of each tile × 180 = $2 ×180 or $360 5. Verify your multiplication. The answer is C. 2. 6 yards Copyright © Houghton Mifflin Company. All rights reserved. SECTION 3.2 Applications and Problem Solving Example 2. Find the cost of building a rectangular driveway measuring 12 feet by 30 feet if concrete costs $12.50 per square yard. A. $4500 B. $400 C. $1500 D. $500 147 Solution Since the cost is per sq. yd, we change 12 ft to 4 yd and 30 ft to 10 yd. In square yards, the area of the driveway is: 4 yd × 10 yd = 40 sq. yd. Total Cost = Cost per sq. yd × sq. yd needed = $12.50 × 40 = $500 The answer is D. Some applications in the CLAST require the use of the Pythagorean theorem. 1 PYTHAGOREAN THEOREM RULE EXAMPLES For any right triangle, the square of the length A jogger runs 4 miles east and then 3 miles of the hypotenuse is equal to the sum of the north. How far is the jogger from his starting squares of the lengths of the other two sides. point? In symbols, If we let a = 4 and b = 3 in the diagram, we have to find the distance c. By the c2 = a2 + b2 Pythagorean theorem, Note: In a right triangle (a triangle with a 90 c2 = a2 + b2 degree angle) the hypotenuse c is the side 2 = 2 Thus, c 3 + 42 opposite the 90 degree angle. c2 = 9 + 16 c c2 = 25 b c= 5 a Thus, he is 5 miles from the starting point. CLAST EXAMPLE Example 3. A television antenna 12 ft high is to be anchored by three wires each attached to the top of the antenna and to points on the roof 5 ft from the base of the antenna. If wire costs $0.75 per foot, what will be the cost of the wire needed to anchor the antenna? A. $27 B. $29.25 C. $9.75 D. $38.25 Hint: After you read the problem, see if you can draw a picture of the situation. c 12 ft 5 ft Solution 1. Read the problem. The antenna is 12 ft high and each wire is 5 ft from the base. The cost of each foot of wire is $0.75. 2. Select the unknown. We need to find the length of each wire, multiply by 3 (we need 3 wires) and then by $0.75 (each foot of wire is $0.75) 3. Think of a plan. First, find the length of each wire. Draw a picture like the one at left. 4. Use the Pythagorean theorem c2 = 52 + 122 c2 = 25 + 144 c2 = 169 c = 13 Thus, each wire is 13 ft long. Since we need three wires, we need 3 ×13 ft = 39 ft. The total cost would be $0.75 ×39 = $29.25. The answer is B. Copyright © Houghton Mifflin Company. All rights reserved. 148 CHAPTER 3 Geometry and Measurement B. Inferring and Selecting Geometric Formulas Objectives IIIB1, IIIB2 CLAST SAMPLE PROBLEMS 1. In the figure, S represents the sum of the measures of the interior angles. Study the figure and calculate the sum S of the measures of the interior angles of an eight-sided convex polygon. 3 sides 1 triangle S = 180o 2. 4 sides 2 triangles S = 360o 6 sides 4 triangles S = 720o The figure consists of a rectangle and a semi-circle of radius r. Determine the formula for finding the area of the shaded region. Many geometric measurements can be generalized to be used in more complicated situations. Look at the three spheres with radii 1, 2 and 3 units and surface areas (SA) 4 π , 16 π and 36 π , respectively. The CLAST asks us to generalize this information and find the surface area of a sphere with a radius of 4 units. To solve this problem set up a table and see if there is a pattern. Radius 1 2 3 4 Surface Area 4 π = 1•4 π 16 π = 2•8 π 36 π = 3•12 π = 4•? r=1 SA = 4 π r=2 SA = 16 π r=3 SA = 36 π The underlined numbers are 4 π , 8 π and 12 π . The next number in the pattern is 16 π so the surface area of a sphere with radius 4 should be 4•16 π or 64 π . (You can also reason that since the surface area is measured in square units, the pattern should involve r2, the square of the radius, that is 12•? = 4 π , 22•? = 16 π , 32•? = 36 π , where we multiply the square of the radius by 4 π . For a radius of 4 units, the answer should be 42 • 4 π = 64 π .) ANSWERS 1. S = 1080o 2. 2r2 - Copyright © Houghton Mifflin Company. All rights reserved. π r2 2 SECTION 3.2 Applications and Problem Solving 149 CLAST EXAMPLE Example Solution 4. In each of the figures, S represents the shaded area when a triangle is removed from a square Side = 4 S=8 Side = 6 S = 18 Side = 8 S = 32 B. 75 C. 50 The area of the first square is 42 = 16 The area of the second square is 62 = 36 The area of the third square is 82 = 64 Now, from the diagram, S = 8, 18 and 32. What is the relationship between the area of the squares and the area S? Area of squares Area of S Find S if the side of the square is 10 A. 100 First find the areas of each of the squares. D. 40 16 8 36 18 64 32 In each case, S is half of the area of the corresponding square. If the side of a square is 10, its area is 100 and S is half of that or 50. The answer is C. CLAST EXAMPLE Example 5. Study the information given with the regular hexagons. Then calculate the height h of the corresponding triangle in a regular hexagon with each side 4 units long. h h b h b b s = 1 unit s = 2 units 3 h = 2 units h = 3 units s = 3 units 3 3 h = 2 units A. 2 3 units B. 3 3 units C. 4 3 units D. 5 3 2 units Solution Look at the pattern involving the length s of the side of the hexagon and the height h. Length of side Height 3 1 1• 2 3 2 2• 2 = 3 3 3 3 3 3• 2 = 2 In each case multiply the length s of the side by 3 2 . For a hexagon whose side measures 4 3 units, the height is 4• 2 = 2 3 . The answer is A. Note: We were looking for a linear pattern, so we used a linear factor. Copyright © Houghton Mifflin Company. All rights reserved. 150 CHAPTER 3 Geometry and Measurement Some of the patterns appearing in the CLAST involve the measure of an angle which is in degrees. You should know that the sum of the measures of the angles of any triangle is 180o. (Read "180 degrees".) CLAST EXAMPLE Example Solution 6. Study each figure below and then calculate S, the sum of the measures of the interior angles of a 9-sided convex polygon. 4 sides 2 triangles S = 360o 5 sides 3 triangles S = 540o 6 sides 4 triangles S = 720o A. 14400 B. 1260o C. 1080o D. 1000o We want to discover a pattern involving the number of sides, the number of triangles and S, the sum of the measures of the angles. No. of sides No. of triangles S o 4 2 2•180 = 360o 5 3 3•180o = 540o 6 4 4•180o = 720o 9 ? ?•180o Note that the number of triangles is 2 less than the number of sides. If the number of sides is 9, the number of triangles is 9 - 2, or 7, and S = 7•180o = 1260o. The answer is B. So far we have inferred new formulas for different geometric figures. A different CLAST skill requires you to calculate the measure of a geometric figure composed of several figures for which the formulas are known. Here is the rule you need. 2 FORMULAS FOR AREAS AND VOLUMES RULE EXAMPLES Study the figure of a right circular cylinder 1. Write the formulas necessary to find the with a hemisphere (half a sphere) mounted on area or volume of the individual figures. top. Find the volume. 2. Use the figure to find the variables to be r used in the formula. r r 3. Simplify the area or volume formula. 4. Find the sum of the areas or volumes. 5. If the area (or volume) to be found is shaded, you may be able to find it by calculating the difference between the areas or volumes. (See Problems 16, 30 and 31.) h h 4 The volume of the sphere is 3 π r3. One half 2 1 4 of this is 2 × 3 π r3 = 3 π r3 The volume of the cylinder is π r2h. Thus the total volume is 2 the indicated sum 3 π r3 + π r2h. Copyright © Houghton Mifflin Company. All rights reserved. SECTION 3.2 Applications and Problem Solving 151 CLAST EXAMPLE Example Solution 7. The figure shows a regular hexagon. Select the formula for computing the total area of the hexagon. The total area of the hexagon is composed of 6 identical triangles as shown. The area h h b b A. Area = 3h + b B. 6(h + b) C. 6hb D. 3hb 1 of each triangle is 2 bh. Since there are 6 of 1 them, the total area is 6•2 bh = 3bh. Since 3bh = 3hb, the answer is D. Section 3.2 Exercises WARM-UPS A 1. Find the cost of tiling a room measuring 12 feet by 18 feet if square tiles costing $2 each and measuring 8 in. by 8 in. are used. 2. A gardener wants to prepare a flower bed measuring 10 feet by 9 feet and 6 inches deep. How many cubic yards of soil are needed to do this? 3. Roofing material come in 100 sq. ft. bundles costing $90 each. How much would the material cost to repair a roof measuring 15 ft by 20 ft? 4. How many square yards of wallpaper costing $10 per roll are needed to cover an area 10 ft long and 8 ft high? 5. A baking pan in the shape of a rectangular solid is 20 cm long, 10 cm wide and 4 cm deep. How many liters will it hold? Hint: 1 liter = 1000 cc. 6. What will be the cost of carpeting an office measuring 12 ft by 18 ft if carpeting costs $15 per square yard? 7. How much would it cost to fence a 30 ft by 60 ft yard if fencing material costs $12 per linear foot? 8. A paper cup is in the shape of a right circular cone 10 cm high and 6 cm in diameter. How many cubic centimeters does the cup hold? Copyright © Houghton Mifflin Company. All rights reserved. 152 CHAPTER 3 Geometry and Measurement 9. A radio station is going to construct a 16 foot antenna anchored by three cables, each attached to the top of the antenna and to points on the roof of the building that are 12 feet from the base of the antenna. What is the total length of the three cables? 10. An airplane leaves the airport and flies 40 miles north. It then flies 30 miles west. How far from the airport is the plane? 11. The door in a warehouse is 9 ft tall and 48 inches wide. If a sheet of paneling is 10 ft long, what is the widest it can be and still fit through the door? 12. A contractor wants to put a sidewalk in front of the triangular garden shown. If it costs $50 per linear foot to build the sidewalk, how much will the job cost? CLAST PRACTICE A Front 8 yd 10 yd PRACTICE PROBLEMS: Chapter 3, # 12-15 13. A farmer wants to build a fence to enclose a rectangular area 20 ft long by 12 ft wide. many feet of fencing will be needed? A. 30 ft B. 23 ft C. 64 ft How D. 112 ft 14. Find the cost of carpeting an office that measures 21 ft by 24 ft if carpet costs $13 per square yard. A. $6552 B. $2184 C. $1092 D. $728 15. The outside dimensions of a picture frame are 4 ft by 30 inches. If its inside dimensions are 3 24 ft by 21 inches, find the area of the frame. A. 759 sq. in. B. 747 sq. in. C. 67.50 sq. ft D. 182.25 sq. ft 16. A gardener uses a six-inch mulch border around a garden measuring 6 ft in diameter. What is the area of the border? A. C. 3.2 sq. in. 2.75 π sq. ft B. D. 6 inches 6 feet 5 π sq. ft 2.75 sq. ft 17. The city commission wants to construct a new street that connects Main Street and North Boulevard as shown in the diagram. If the construction cost is $90 per linear foot, find the estimated cost for constructing the street. Hint: 1 mile = 5280 ft. N New 4 mi North Blvd 3 miles Main St A. $33,808 B. $3690 C. $2,376,000 D. $475,200 Copyright © Houghton Mifflin Company. All rights reserved. E S SECTION 3.2 Applications and Problem Solving 153 WARM-UPS B 18. Study the information given with the right triangles then calculate the area of a right triangle with hypotenuse 10 and base 5 3 . 4 6 8 3 3 4 3 2 2 3 3 1 A = 2 •2 3 •2 1 A = 2 • 3 •1 19. 1 A = 2 •4 3 • Study the given figures then find the area of a the regular 6- sided polygon. h h h b 4 sides A = 2 bh 20. 1 A = 2 •3 3 •3 b b 5 sides 5 A = 2 bh 6 sides A=? Study the figure formed from a semicircle and a triangle. Then give the formula for calculating the total area of the figure. r h 21. Study the figure showing a square with a semicircle attached to each side. Find the formula for computing the perimeter of the figure. 22. Find the formula for the total area of the figure in Problem 21. s Copyright © Houghton Mifflin Company. All rights reserved. 154 CHAPTER 3 Geometry and Measurement CLAST PRACTICE B 23. Study the information given with the regular hexagons, then calculate the area A of a regular hexagon with a side equal to 8. A. 96 3 C. 24 25 48 3 B. 72 3 D. 36 3 h h h b b Side 2 A=6 3 Side 4 A = 24 3 b Side 6 A = 54 3 Study the information given for the 3, 4 and 6 sided figures. If S represents the sum of the measures of the interior angles, find S for a 10 sided polygon. A. 1080o B. 1800o C. 1440o D. 1260o 3 sides 1 triangle S = 180o 4 sides 2 triangles S = 360o The given figure is formed by joining a triangle and a square as shown. Study the figure then select the formula for computing its total area. A. A = s4 3s2 C. A = 4 + s2 26. PRACTICE PROBLEMS: Chapter 3, #16-20 s A. SA = 2(2 π r) + rh B. SA = π r2h C. SA = 2 π r2 + 2 π rh D. SA = 2 π r2 + 2rh Copyright © Houghton Mifflin Company. All rights reserved. s s B. A = s3 5 s2 D. A = 4 A cylindrical container can be formed from two tops and a rectangle as shown. Based on the figure, what is the formula for calculating the surface area SA of the right circular cylinder? 6 sides 4 triangles S = 720o s r r h r h SECTION 3.2 Applications and Problem Solving 27. The figure shows a rectangular box without a top formed by cutting out squares x units on each side from a rectangular piece of length L and width W. Select the formula for calculating the surface area SA of the box. A. B. C. D. 28. 29. 30. 155 x x W SA = LW - 4x2 SA = LW - x2 SA = (L - 2x)W - 2x) - 4x2 SA = (L - 2x)(W - 2x) L Study the figure showing a regular pentagon. Then select the formula for computing the total area A of the pentagon. A. 5 A = 2 hb B. A = 5(h + b) C. A = 5h + b D. A = 5hb Study the figure showing a trapezoid, then select the formula for computing the total area A of the trapezoid, if B = x + b + y. A. A = b(B - x - y) + xh2 B. A = h(B + b) C. A = b(B - x - y)h 1 D. A = 2 h(B + b) Study the figure showing a square inscribed in a circle. Select the formula for computing the area of the shaded region. A. A = π r2 - s2 s2 B. A = π 2 - s2 s2 C. A = 2 s2 D. A = π 2 h b h h b y x s Copyright © Houghton Mifflin Company. All rights reserved. 156 31. CHAPTER 3 Geometry and Measurement Study the figure showing a triangle inscribed in a circle. Select the formula for computing the area of the shaded region. A. A = π r2 - 2r2 B. A= π -2 C. A= π D. A = π r2 - r2 r EXTRA CLAST PRACTICE 32. Study the information given in the figures and then calculate A, the exterior angle in the last figure. A. 65o B. C. 110o o 60 135o D. 115o o 70 o 130 o 20 100 45 o 25 90 o o 115 o 65 120 o o o A 33. The base of a tool shed is to be a slab of concrete 15 feet long by 12 feet wide by 6 inches thick. If one cubic yard of concrete costs $39, how much will the concrete for the base of the tool shed cost? A. $3510 B. $1560 C. $130 D. $65 1 34. A rectangular flower bed measures 22 feet by 50 inches. The outside dimensions of a path around the bed are 4 ft by 60 inches. What is the area of the path? A. 115 square feet B. 200.50 square feet C. 1200 square inches D. 1380 square inches 35. The diameter of a tree trunk is 2.4 meters. What is its circumference? A. 0.72 π square meters B. 1.2 π meters C. 2.4 π meters D. 2.88 π square meters Copyright © Houghton Mifflin Company. All rights reserved. SECTION 3.3 Lines, Angles and Triangles 157 3.3 LINES, ANGLES AND TRIANGLES Before we proceed to discuss the different types of lines, angles and triangles we need to learn some of the basic terminology. T TERMINOLOGY--TYPES OF ANGLES PLANE ANGLES EXAMPLES A plane angle is a figure formed by two rays C with a common endpoint called the vertex. vertex The rays AB and AC are the sides of the angle 1 A and the angle can be named ∠ 1 (read "angle B 1"), ∠ CAB , ∠ BAC or ∠ A The vertex of angle BAC is A. Angles are measured by the amount of rotation needed to turn one side of an angle so that it coincides with (falls exactly on top of) the other side. Note that ∠ YXZ. is "greater than" ∠ BAC. The rotation of an angle is measured in degrees and is denoted by m ∠ BAC. One complete revolution is 360o (read "360 1 degrees") and 360 of a complete revolution is 1o. The CLAST sometimes states that the measure of ∠ A, for example, is represented by "x". C Z A X B A complete revolution is 360o C C A 30° x A B B m ∠ BAC = 30o One-half of a complete revolution is 180o and results in an angle called a straight angle. One-quarter of a complete revolution is 90o and results in an angle called a right angle (denoted by using the little square ). Y x = 30o 180° 90° An acute angle is an angle whose measure is less than 90o. An obtuse angle is an angle whose measure is greater than 90o and less than 180o. Copyright © Houghton Mifflin Company. All rights reserved. 158 CHAPTER 3 Geometry and Measurement We now give several relationships between angles. T TERMINOLOGY--COMPLEMENTARY, SUPPLEMENTARY, VERTICAL ANGLES COMPLEMENTARY ANGLES EXAMPLES o Two angles whose measures add to 90 are called complementary angles. B Note that A m ∠ A + m ∠ B = 90o or m ∠ A = 90o - m ∠ B SUPPLEMENTARY ANGLES Two angles whose measures add to 180o are B A supplementary angles. Note that o o m ∠ A + m ∠ B = 180 or m ∠ A =180 - m ∠ B VERTICAL ANGLES When two lines intersect, the opposite angles are called vertical angles and they have the u x y same measure. If x, y, u and v are the v measures then x = y and u = v. We are now ready to study the properties existing between different plane figures such as angles and triangles. Before we do that, let us see what type of questions we will encounter. A. Properties of Plane Figures Objective IIB2 CLAST SAMPLE PROBLEMS 1. What type of triangle is Δ ABC below? A 65 o B 45 o C In the figure to the right: 2. 3. 4. 5. Find two right angles Find two pairs of complementary angles Find two pairs of vertical angles that are not right angles Which geometric figure has all of the following characteristics? i. quadrilateral ii. opposite sides are parallel iii. diagonals are not equal ANSWERS 1. Scalene 4. 3 and 6; 2 and 5 2. 1 and 4 5. Rhombus Copyright © Houghton Mifflin Company. All rights reserved. 3 2 4 1 5 6 3. 5 and 6; 2 and 3 SECTION 3.3 Lines, Angles and Triangles 159 CLAST EXAMPLE Example Solution Complementary angles are angles whose sum is 90o, so 1 can not be complementary to any other angle. (Eliminate answers A and D.) Now, 5 and 6 are complementary, but this is not one of the choices. However, angles 6 and 3 are vertical angles, so they must have the same measure. Since angles 5 and 6 are complementary and angles 3 and 6 have equal measures, ∠ 5 and ∠ 3 are complementary. The answer is C. If you are not convinced, here is the proof: 1. 6 1 5 4 2 3 Which of the following pairs of angles are complementary? A. 1 and 2 B. 5 and 2 C. 3 and 5 D. 4 and 1 m ∠ 5 + m ∠ 6 = 90o m∠ 6 = m∠ 3 m ∠ 5 + m ∠ 3 = 90o, so ∠ 3 and ∠ 5 are complementary. Triangles can be classified according to their angles (right, acute or obtuse) or according to the number of equal sides (scalene, isosceles, or equilateral). T TERMINOLOGY -- CLASSIFYING TRIANGLES BY ANGLES AND SIDES CLASSIFICATION EXAMPLES A right triangle is a triangle containing a right angle. An acute triangle is a triangle in which all the angles are acute. An obtuse triangle is a triangle containing an obtuse angle. A scalene triangle is a triangle with no equal sides. Note that the sides are labeled |, || and ||| to indicate that the lengths of the sides are different. Copyright © Houghton Mifflin Company. All rights reserved. 160 T CHAPTER 3 Geometry and Measurement TERMINOLOGY--CLASSIFYING TRIANGLES BY ANGLES AND SIDES (CONT.) An isosceles triangle is a triangle with two equal sides and two equal base angles. Note that the equal angles are indicated by arcs, the equal sides by slashes. An equilateral triangle is a triangle with three equal sides and three 60o angles. As before the equal angles are indicated by arcs and the equal sides by slashes. Next, we give the rules used to name specific triangles, and a very important rule that applies to the measures of the sum of the angles of any type of triangle. Do you remember what that is? T TERMINOLOGY--NAMING TRIANGLES NAMING TRIANGLES EXAMPLES Triangles are named using the letters at the A vertices and the symbol Δ . The given triangle is Δ ABC (read "triangle ABC"). C B 1 SUM OF THE ANGLES OF A TRIANGLE RULE 60o 35 o The sum of the measures of the angles of a triangle is 180o. You can use this fact to find The measure of the other angle, call it x, is the measure of the third angle when the measures of the other two angles are given. such that: 60o + 35o + x = 180o or 95o + x = 180o x = 85o CLAST EXAMPLE Example 2. What type of triangle is Δ ABC? A 55 o 70 o B C A. Isosceles C. Equilateral B. D. Right Scalene Solution First, we have to determine the measure "x" of the missing angle. Since the sum of the three angles in a triangle is 180o, we have: 55o + 70o + x = 180o 125o + x = 180o x = 180o - 125o x = 55o Since two of the angles are of equal measure, the triangle is an isosceles triangle. Thus, A. Copyright © Houghton Mifflin Company. All rights reserved. SECTION 3.3 Lines, Angles and Triangles 161 In the preceding sections we have mentioned geometric figures such as squares, circles, rectangles and polygons. These figures have definite properties that should be familiar to you. We shall give the definitions and examples for several polygons and quadrilaterals. T TERMINOLOGY--TYPES OF POLYGONS TYPES OF POLYGONS EXAMPLES A polygon is a closed figure whose sides are line segments that do not cross each other. A regular polygon is a polygon with all sides of equal length and all angles of equal measure. They are usually named by using the number of sides. For example: Pentagon is a 5-sided polygon. Hexagon is a 6-sided polygon. Octagon is an 8-sided polygon. A F E A B B E D C D C A quadrilateral is a four-sided polygon (Squares and rectangles are quadrilaterals). A trapezoid is a quadrilateral with one pair of parallel sides. A parallelogram is a quadrilateral in which the opposite sides are parallel. A rhombus is a parallelogram with all sides equal. A rectangle is a parallelogram with a right angle. A square is a rectangle with all sides equal. A diagonal is a line segment joining two nonconsecutive vertices of a polygon. Copyright © Houghton Mifflin Company. All rights reserved. 162 CHAPTER 3 Geometry and Measurement The CLAST emphasizes the characteristics of these polygons as shown in the example. CLAST EXAMPLE Example 3. Select the geometric figure that possesses all of the following characteristics: i. polygon ii. quadrilateral iii. two and only two sides are parallel. A. Parallelogram B. Rectangle C. Rhombus D. Trapezoid Solution First note that all of the responses give figures that are polygons and quadrilaterals, so we have to concentrate on the figure having two and only two parallel sides. The parallelogram, the rectangle and the rhombus have more than two parallel sides. The only quadrilateral with exactly one pair of parallel sides is the trapezoid. The answer is D. B. Relationships Between Angle Measures Objective IIB1 CLAST SAMPLE PROBLEMS Referring to the diagram on the right m ∠ g = 130o: 1. Which angles are supplements of g? 2. Find m ∠ f, m ∠ h and m ∠ j 3. Find three angles with the same measure as m ∠ c 4. Find two pairs of supplementary angles that are not right angles. a b d c f g e h j The material we have studied can be used to find additional relations between angles. One of these relations is congruence as detailed next. T TERMINOLOGY--CONGRUENT ANGLES AND SIDES CONGRUENT ANGLES EXAMPLES B Angles ∠ 1 and ∠ 2 are congruent, denoted by ∠ 1 ≅ ∠ 2, means that m ∠ 1 = m ∠ 2. 2 The congruence symbol should remind you of "=". 1 3 A C The angles of an equilateral triangle are congruent, that is, ∠ 1 ≅ ∠ 2 ≅ ∠ 3 CONGRUENT SIDES In the equilateral triangle above Two sides of a triangle are congruent, denoted ___ ___ ___ ___ ___ ___ AB ≅ B C ≅ AC by AB ≅ AC means that side AB is the ___ same length as side AC . ANSWERS 1. f and j 2. 50o, 130o, 50o 3. b, d, e Copyright © Houghton Mifflin Company. All rights reserved. 4. f and g, g and j, j and h, f and h SECTION 3.3 Lines, Angles and Triangles 163 CLAST EXAMPLE Example ___ Solution ___ 4. In Δ ABC, AB ≅ CB . Which of the following statements is true for the figure shown? (The measures of the angles are represented by x, y, z as indicated.) ___ ___ ___ AB ≅ CB means that the line segments AB ___ and CB are the same length. This makes the triangle an isosceles triangle, so y = x. B Now, x + 100o = 180o and x = 80o z y x C A Since y = x = 80o, y = 80o 100° w The correct answer is C. A. z = 80o B. z=x C. y = 80o D. z = 10o The CLAST combines the idea of perpendicular and parallel lines to establish other relationships between the measures of angles. Here are the concepts we need. T TERMINOLOGY--PARALLEL AND PERPENDICULAR LINES PARALLEL LINES EXAMPLES ___ ___ A B Lines AB and CD are parallel, denoted by ___ ___ ___ ___ E AB || CD , if the lines AB and CD are in the same plane and do not intersect.. C D F These pairs of parallel lines are denoted by ___ ___ ___ AB || CD and ___ EF || GH . PERPENDICULAR LINES ___ ___ D Lines AB and CD are perpendicular, ___ ___ denoted by AB ⊥ CD , if they intersect at right angles. A ___ C B ___ Lines AB and CD are perpendicular. Copyright © Houghton Mifflin Company. All rights reserved. G H 164 CHAPTER 3 Geometry and Measurement CLAST EXAMPLE Example ___ ___ Solution ___ ___ 5. Given that CD || BE and AE ⊥ CD which of the following statements is true for the figure at the right? (The measure of angle DCB is represented by "v".) C B A. w = x D. ∠ DCB and ∠ DEB are supplementary angles w o u E y z 135 x B. y = x C. ∠ EDC and ∠ DEB are vertical angles D v A Since y and the 135o angle are supplementary, y = 45o. Now, z = 90o and x + y + z = 180o. For y = 45o, z = 90o, x + 45o + 90o = 180o x + 135o = 180o x = 45o o Since both x and y are 45 , y = x and the correct answer is B. There is one more important fact concerning parallel lines and the measures of related angles. Here it is: T TERMINOLOGY--CORRESPONDING, ALTERNATE AND VERTICAL ANGLES ANGLE RELATIONSHIPS EXAMPLES If L1 and L2 are parallel lines crossed by a 1 2 transversal as shown in the Example, the L1 3 4 following relationships hold: Corresponding angles are congruent. ∠ 1 ≅ ∠ 5, ∠ 3 ≅ ∠ 7, ∠ 2 ≅ ∠ 6 and ∠4 ≅ ∠8 These are the pairs of angles that lie on the same side of the transversal. Alternate interior angles are congruent. ∠ 3 ≅ ∠ 6, and ∠ 4 ≅ ∠ 5. These are the pairs of angles between the parallel lines and on opposite sides of the transversal Vertical angles are congruent. ∠ 1 ≅ ∠ 4, ∠ 2 ≅ ∠ 3, ∠ 5 ≅ ∠ 8, and ∠6≅ ∠7 L2 5 6 7 8 Here is a quick way of remembering all these facts: Look at angles 1 and 2. Angle 1 is acute and 2 is obtuse. Think of ∠ 1 as “small" and ∠ 2 as "big". All acute ("small" ) angles are congruent. Thus, ∠ 1, ∠ 4, ∠ 5, and ∠ 8 are congruent. All obtuse ("big") angles are congruent Thus, ∠ 2, ∠ 3, ∠ 6 and ∠ 7 are congruent. Copyright © Houghton Mifflin Company. All rights reserved. SECTION 3.3 Lines, Angles and Triangles 165 CLAST EXAMPLES Example Solution 6. Which statement is true for the figure shown at right given that L1 and L2 are parallel lines? A. Since m ∠ T = 750, m ∠ S = 60o L2 B. Since m ∠ T = 750, m ∠ S = 105o P L1 Q R 75° S U T W X Y V 45° Look at the transversal on the left m ∠ T = 75o (Remember, all the "small" angles are congruent). Now, ∠ S and ∠ T are suppplementary, so m ∠ S = 180o - m ∠ T = 180o - 75o = 105o. This is answer B. C. m ∠ V = m ∠ R D. None of the statements is true. Example Solution x = 40o because x and the 40o angle are corresponding angles. 7. If L1 || L2 and L3 || L4, which of the following statements is true? y = 180o - 40o = 140o, since x and y are supplementary. s o 40 t r z x L y 3 The correct answer is C. L L L 1 z = y = 140o, since z and y are alternate interior angles (remember "big" angles are congruent) A. z = 40o B. r = 140o C. z = 140o D. s=t 4 2 Copyright © Houghton Mifflin Company. All rights reserved. 166 CHAPTER 3 Geometry and Measurement C. Similar Triangles Objective IIB3 CLAST SAMPLE PROBLEMS A In the figure to the right, the measure of angle A is x CB ? BA ? 1. x = ? . 2. AB = AD 3. BD = ? 4. If the measure of angle A is z, what is AC ? w o 30 x y C o 30 4 E C y 120 o E 120o D z B A z D 5 x 7.5 B Geometric figures with exactly the same shape, but not necessarily the same size, are called similar figures. If you know that two triangles are similar, there are several relationships you should know. We list them next. T TERMINOLOGY--SIMILAR TRIANGLES DEFINITION OF SIMILAR TRIANGLES EXAMPLES E When two triangles are similar, denoted by Δ ABC ~ Δ DEF (1) Corresponding angles are equal and 8 B (2) Corresponding sides are proportional. 4 To remember the information in (1) note: A Δ ABC ~ Δ DEF ∠A ≅ ∠D, ∠B ≅ ∠E, ∠C ≅ ∠ F. Match first and first (A and D), second and second (B and E), third and third (C and F). To remember the information in (2), note: Δ ABC ~ Δ DEF ___ ___ match AB and DE . ___ ___ ___ ___ Δ ABC ~ Δ DEF match BC and EF . Δ ABC ~ Δ DEF match AC and DF . ANSWERS 1. y 4 2 3 C D 6 F The two triangles above are similar. (1) ∠ A ≅ ∠ D, ∠ B ≅ ∠ E, ∠ C ≅ ∠ F, ___ ___ ___ AB BC CA 2 4 3 = ___ = ___ = 4 = 8 = 6 DE EF FD A In the figure at right, if BC is parallel to DE C B Δ ABC ~ Δ ADE D E (2) ___ 2. CE Copyright © Houghton Mifflin Company. All rights reserved. BC 3. BE 4. AC = 6 SECTION 3.3 Lines, Angles and Triangles 167 The CLAST asks two types of questions about similar triangles: 1. From a group of triangles, select the ones that are similar 2. Find specific measures for angles or sides of one of two similar triangles. ÜCLAST EXAMPLES Example Solution To find the correct answer, we have to check each of the responses. The first triangle in A is an equilateral triangle. For the second triangle to be similar, it has to be equilateral and the third angle must also be 55o. But 55o + 55o + 55o = 165o and not 180o, so the second triangle is not equilateral. A is not the answer. 8. Which triangles are similar: A. 5 5 55° 55° 5 B. If the triangles are similar, the sides must be 6 5 proportional. Thus, 9 = 7 . But this is not true, so the triangles are not similar. 9 5 30° 7 6 The two triangles in C are obviously not similar since corresponding angles are not equal. C. A o 25 B D C 60 o E D. Q A 80° 80° 45° 55° B C R S The correct answer must be D. Let us see why. The measure of the angles in the first triangle are 55o and 80o, so the measure of the third angle must be 180o - 80o - 55o = 45o. But the measures of the angle in the second triangle are 80o and 45o, so the third angle must be 55o, making the corresponding angles equal and the triangles similar. Copyright © Houghton Mifflin Company. All rights reserved. 168 CHAPTER 3 Geometry and Measurement Example Solution Since ∠ 1 and ∠ 2 are vertical angles, they have the same measure, so i is true. 9. Which statements are true for the pictured triangles? B C 1 A 5 Triangle BAC and triangle DEC have equal corresponding angles (the 90o angle and angles 1 and 2, which make the third angles equal). Thus, Δ BAC ~ Δ DEC which makes statement ii true. E 2 3 D ___ ___ ___ AB AC ___ i. m∠ 1 = m∠ 2 ii. ___ ED ___ ___ iii. AB is parallel to ED = ___ EC A. i only B. ii only C. i and ii only D. i, ii, and iii ___ Solution Since m ∠ G = m ∠ H (vertical angles) and m ∠ D = m ∠ B = 40o, the third angles must be equal. Thus, m ∠ A = m ∠ E, and i is true. A 7.5 5 B C 4 E i. iii. m∠ A = m∠ E ___ ___ CE CB = ___ CD ___ CA Since Δ CDE ~ Δ ABC, corresponding sides are proportional. Thus, 40° H G ___ This makes iii a true statement and the correct answer is D. 10. Which statements are true for the pictured triangles? 40° ___ AB ⊥ BD and DE ⊥ BD . Example D ___ Finally, AB is parallel to ED because ___ ii. AC = 6 ___ ___ 4 AC 4•7.5 30 5 = 7.5 or AC = 5 = 5 = 6 hence ii is true. Since the two triangles are similar, ___ ___ CE CD ___ CA A. i only B. ii only C. i and ii only D. i, ii, and iii = ___ , so iii is not true. CB The correct answer is C, i and ii are the only true ones. Copyright © Houghton Mifflin Company. All rights reserved. SECTION 3.3 Lines, Angles and Triangles 169 Section 3.3 Exercises WARM-UPS A 1. Name two pairs of complementary angles in the figure. 2. Name a pair of supplementary angles in the figure. 3. Name two pairs of vertical angles in the figure. 4. Name two right angles in the figure. 5. Name two acute angles in the figure. 6. Is there an obtuse angle in the figure? 7. Find m ∠ ABC B 8. Find m ∠ 3 6 5 4 1 2 3 9. Find x 10. Find y and z y A o o 60 60 Triangle 1 C x o 100 50o o 30 3 Triangle 2 Triangle 3 60 o o 75 z Triangle 4 In Exercises 11-14, classify the triangles according to their angles and sides. 11. What type of triangle is triangle 1? 12. What type of triangle is triangle 2? 13. What type of triangle is triangle 3? 14. What type of triangle is triangle 4? 15. Name the geometric figure that is a parallelogram in which the diagonals are perpendicular and of the same length. 16. Name a quadrilateral in which only two sides are parallel. Copyright © Houghton Mifflin Company. All rights reserved. 170 CHAPTER 3 Geometry and Measurement CLAST PRACTICE A 17. PRACTICE PROBLEMS: Chapter 3, # 21-22 Which of the following pairs of angles are supplementary, given that L1 and L2 are parallel lines? A. ∠ P and ∠ S B. ∠ Y and ∠ Q D. ∠ R and ∠ Q D. ∠ R and ∠ X 18. What type of triangle is Δ ABC? A. Isosceles B. Equilateral C. Right D. Obtuse 19. i. X L2 A 40 S Y 50 o o B C B. C. D. Select the geometric figure that possesses all of the following characteristics: Triangle A. C. 21. R Q Which of the angles is an obtuse angle? A. 20. P L1 ii. Scalene triangle Acute triangle All acute angles iii. All sides are equal B. Equilateral triangle D. Isosceles triangle The words "right," "acute," "obtuse," "scalene," "isosceles," and "equilateral" are used individually to describe triangles. Which of the following combinations of words are impossible to describe a triangle? A. C. Right; scalene Right; obtuse B. Isosceles; obtuse D. Isosceles; right Copyright © Houghton Mifflin Company. All rights reserved. SECTION 3.3 Lines, Angles and Triangles 171 WARM-UPS B 22. Find m ∠ 1. 23. Find m ∠ 4. 1 2 L 4 L 3 24. Find m ∠ 2. 5 L 1 ___ 115 o 1 2 and L are parallel 2 ___ 25. In D ABC, AB ≅ CB . Find m ∠ y. B z ___ ___ 26. In D ABC, AB ≅ CB . Find m ∠ x. o 110 y A ___ x C w ___ 27. In D ABC, AB ≅ CB . Find m ∠ z. ___ ___ ___ ___ Given that CD || BE and AD ⊥ CD . 28. Find m ∠ x. 29. Find m ∠ y. 30. Find m ∠ u. 31. Find m ∠ z. C D 45o w B v u E y z x A Copyright © Houghton Mifflin Company. All rights reserved. 172 CHAPTER 3 Geometry and Measurement Given that L1 and L2 are parallel 32. Find m ∠ Q. S L2 CLAST PRACTICE B R 30 75° 33. Find m ∠ T. 34. Find m ∠ W. Q P L1 U o W T X Y V PRACTICE PROBLEMS: Chapter 3, # 23-24 35. Which statement is true for the figure? A. m ∠ B = 100o B. m ∠ E = 20o C. m∠ C ≠ m∠ H D. ∠ A and ∠ B are complementary 36. Which statement is true for the figure? E A. m∠ E π m∠ C B. m ∠ I = 120o C. m∠ B = m∠ G D. ∠ A and ∠ I are supplementary 8 A i only B. ii only C. iii only D. i and iii only 38. Which of the statements is true for the figure shown, given that L1 and L2 are parallel lines? A. m∠ P = m∠ Y B. m ∠ P = 100o C. ∠ X and ∠ P are supplementary D. ∠ Q and ∠ X are complementary B C F 4 D C A B D 100° G H E 6 37. Which statements are true for the figure if M1 and M2 are parallel? i. m ∠ 3 = 50o ii. m ∠ 4 = 70o iii. ∠ 1 and ∠ 2 are supplementary A. 8 6 F I G H 6 70° 1 3 L1 L2 Copyright © Houghton Mifflin Company. All rights reserved. 2 M1 4 50° M2 P Q 80° X Y SECTION 3.3 39. i. ii. iii. The diagram to the right shows lines in the same plane. If line L1 is a horizontal line, which statement is true? L2 and L3 are the only parallel lines L2 and L4 are perpendicular lines Lines L2 and L3 are vertical lines. A. i only B. ii only C. iii only D. i and iii only Lines, Angles and Triangles L2 173 L3 L1 60° 60° L4 L5 40. Given that L1 || L2, which of the following statements is true? A. u + v = 180o C. u = 40o ___ B. D. ___ C AC ⊥ BC u y = 50o o 40 y B 130 o A L v x z 1 L 2 WARM-UPS C 41. The two triangles to the right are similar. Find the lengths marked x and y. 5 y 4 x 3 4 42. The parallelograms to the right are similar. __ Find the length of the diagonal PR . S D R C 10 10 A 8 B P 10 Q Copyright © Houghton Mifflin Company. All rights reserved. 174 CHAPTER 3 Geometry and Measurement __ In the figure to the right, the line PQ is parallel to the line AB In each problem, find the missing lengths. __ __ ___ __ AP PC BQ BC 43. 3 4 6 ? 44. 5 4 ? 6 45. 2 ? 3 5 46. ? 4 4 8 C Q P B A CLAST PRACTICE C PRACTICE PROBLEMS: Chapter 3, # 25-27 47. Study figures A, B, C, D, then select the figure in which all triangles are similar. B. A. 80° 80° 75° 70° C.. 48. D. Study figures A, B, C, D, then select the figure in which all triangles are similar. A. B. C. D. 5 60° 5 60° 5 5 58° 9 30° 7 58° Copyright © Houghton Mifflin Company. All rights reserved. 6 SECTION 3.3 49. A. Which statement is true for the triangles to the right? m∠ A = m∠ E B. m ∠ x = 30o ___ C. AC = 3.5 ___ ___ CE CB D. ___ CA C. m∠ X = m∠ Z ___ ___ CE AB ___ AD = ___ CB z D 30° x 4 30° 3 E A X C Y 130° E D Z EXTRA CLAST PRACTICE ___ 51. ___ Given that AB || CD and ___ ___ BD ⊥ CD which of the following statements is true for the figure shown. (The measure of angle ACD is represented by x) 52. E A A. w=v B. ∠ EAB and ∠ EBA are supplementary C. ∠ AEB and ∠ ECD are complementary angles D. z=x C x v u z B y D w Select the geometric figure that possesses all of the following characteristics i. four sided ii. right angle iii. adjacent sides equal and opposite sides parallel A. rectangle B. square C. trapezoid D. rhombus B C 130° None is true 5 y B. m ∠ X = m ∠ Y D. 175 A = ___ CD 50. Which statement is true for the triangles to the right? A. Lines, Angles and Triangles Copyright © Houghton Mifflin Company. All rights reserved. B

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