DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A Name 1.1 Class Date Segment Length and Midpoints Essential Question: How do you draw a segment and measure its length? Resource Locker Explore Exploring Basic Geometric Terms In geometry, some of the names of figures and other terms will already be familiar from everyday life. For example, a ray like a beam of light from a spotlight is both a familiar word and a geometric figure with a mathematical definition. The most basic figures in geometry are undefined terms, which cannot be defined using other figures. The terms point, line, and plane are undefined terms. Although they do not have formal definitions, they can be described as shown in the table. © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Marco Vacca/age fotostock Undefined Terms Term Geometric Figure A point is a specific location. It has no dimension and is represented by a dot. P A line is a connected straight path. It has no thickness and it continues forever in both directions. A plane is a flat surface. It has no thickness and it extends forever in all directions. B A point P ℓ X line ℓ, line AB, line BA, ‹ › − ‹ › − AB, or BA Z Ways to Name the Figure Y plane or plane XYZ In geometry, the word between is another undefined term, but its meaning is understood from its use in everyday language. You can use undefined terms as building blocks to write definitions for defined terms, as shown in the table. Defined Terms Term Geometric Figure A line segment (or segment) is a portion of a line consisting of two points (called endpoints) and all points between them. A ray is a portion of a line that starts at a point (the endpoint) and continues forever in one direction. Module 1 GE_MNLESE385795_U1M01L1.indd 5 C D P Q 5 Ways to Name the Figure segment _ _CD, segment DC, CD, or DC → ‾ ray PQ or PQ Lesson 1 3/20/14 5:03 PM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A DO NOT Correcti You can use points to sketch lines, segments, rays, and planes. A B Draw two points J and K. Then draw a line through them. (Remember that a line shows arrows at both ends.) Draw two points J and K again. This time, draw the line segment with endpoints J and K. J J K K C D Draw a point K again and draw a ray from endpoint K. Plot a point J along the ray. Draw three points J, K, and M so that they are not all on the same line. Then draw the plane that contains the three points. (You might also put a script letter such as on your plane.) J J M K E K Give a name for each of the figures you drew. Then use a circle to choose whether the type of figure is an undefined term or a defined term. Point Line Segment Plane undefined term/defined term ‹ › ‹ › − − JK (or KJ ) undefined term/defined term _ _ JK or KJ undefined term/defined term → ‾ KJ undefined term/defined term plane JKM (or plane ) undefined term/defined term Reflect 1. 2. → → ‾ be the same ray as KJ ‾ ? Why or why not? In Step C, would JK No. The rays would have different endpoints and continue in opposite directions. In Step D, when you name a plane using 3 letters, does the order of the letters matter? No. Using 3 letters, the plane in Step D can be named plane JKM, plane JMK, plane KJM, © Houghton Mifflin Harcourt Publishing Company Ray points J, K, and M plane KMJ, plane MJK, or plane MKJ. 3. ‹ › ‹ › − − Discussion If PQ and RS are different names for the same line, what must be true about points P, Q, R, and S? The four points all lie on a common line. Module 1 GE_MNLESE385795_U1M01L1.indd 6 6 Lesson 1 3/20/14 5:02 PM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A Constructing a Copy of a Line Segment Explain 1 The distance along a line is undefined until a unit distance, such as 1 inch or 1 centimeter, is chosen. You can use a ruler to find the distance between two points on a line. The distance is the absolute value of the difference of the numbers on the ruler that 0cm correspond to the two points. This distance is the length of the segment determined by the points. _ In the figure, the length of RS, written RS (or SR), is the distance between R and S. RS = ⎜4 - 1⎟ = ⎜3⎟ = 3 cm or S R 1 2 3 4 5 SR = ⎜1 - 4⎟ = ⎜-3⎟ = 3 cm Points that lie in the same plane are coplanar. Lines that lie in the same plane but do not intersect are parallel. Points that lie on the same line are collinear. The Segment Addition Postulate is a statement about collinear points. A postulate is a statement that is accepted as true without proof. Like undefined terms, postulates are building blocks of geometry. Postulate 1: Segment Addition Postulate Let A, B, and C be collinear points. If B is between A and C, then AB + BC = AC. A B C A construction is a geometric drawing that produces an accurate representation without using numbers or measures. One type of construction uses only a compass and straightedge. You can construct a line segment whose length is equal to that of a given segment using these tools along with the Segment Addition Postulate. Example 1 Use a compass and straightedge to construct a segment whose length is AB + CD. A B D C © Houghton Mifflin Harcourt Publishing Company Step 1 Use the straightedge to draw a long line segment. Label an endpoint X. (See the art drawn in Step 4.) Step 2 To copy segment AB, open the compass to the distance AB. A Step3 Place the compass point on X, and draw an arc. Label the point Y where the arc and the segment intersect. Step4 To copy segment CD, open the compass to the distance CD. Place the compass point on Y, and draw an arc. Label the pointZ where this second arc and the segment intersect. X Y B Z _ XZ is the required segment. Module1 GE_MNLESE385795_U1M01L1.indd 7 7 Lesson1 3/20/14 5:02 PM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A B DO NOT Correcti B A C D Step 1 Use the straightedge to draw a long line segment. Label an endpoint X. Step 2 To copy segment AB, open the compass to the distance AB. Step 3 Place the compass point on X, and draw an arc. Label the point Y where the arc and the segment intersect. Step 4 To copy segment CD, open the compass to the distance CD. Place the compass point on Y, and draw an arc. Label the point Z where this second arc and the segment intersect. _ XZ is the required segment. X Y Z Reflect 4. Discussion Look at the line and ruler above Example 1. Why does it not matter whether you find the distance from R to S or the distance from S to R? The formula to find the distance between the two points involves taking the absolute value of the difference between the two coordinates R and S, so the distance is always positive; the order of the coordinates does not matter. From R to S or from S to R, the coordinates are always 3 units apart. 5. _ _ In Part B, how can you check _length of YZ is the same as the length of CD? _ that the Use a ruler to measure YZ and CD to see if the lengths are the same. Your Turn © Houghton Mifflin Harcourt Publishing Company 6. Use a ruler to draw a segment PQ that is 2 inches _ long. Then use your compass and straightedge to construct a segment MN with the same length as PQ. P Q M Explain 2 N Using the Distance Formula on the Coordinate Plane The Pythagorean Theorem states that a 2 + b 2 = c2, where a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse. You can use the Distance Formula to apply the Pythagorean Theorem to find the distance between points on the coordinate plane. Module 1 GE_MNLESE385795_U1M01L1.indd 8 8 Lesson 1 3/20/14 5:02 PM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A The Distance Formula The distance between two points (x 1, y 1) and (x 2, y 2) on the coordinate plane is y ―――――――― √(x 2 - x 1) 2 + (y 2 - y 1) 2. (x2, y2) (x1, y1) x Example 2 Determine whether the given segments have the same length. Justify your answer. y 4 A C E B -4 0 x H 4 D F G -4 _ _ AB and CD Write the coordinates of the endpoints. _ Find the length of AB. Simplify the expression. _ Find the length of CD. Simplify the expression. ― A(-4,4), B(1,2), C(2,3), D(4, −2) ―――――――― (2 - 4) ―――― 29 = √5 + (-2) = √― ―――――――― CD = √(4 - 2) + (-2 - 3) ―――― 29 = √2 + (-5) = √― AB = √(1 - (-4)) + 2 2 2 2 2 2 2 2 © Houghton Mifflin Harcourt Publishing Company _ _ So, AB = CD = √29 . Therefore,AB and CD have the same length. _ _ EF and GH Write the coordinates of the endpoints. Find the length of _ EF. Find the length of ) ――――――――――― 2 2 EF = = Simplify the expression. _ GH. ( E(-3, 2), F -2 , -3 , G(-2, -4), H GH = = Simplify the expression. √( ) ( ) (2 , 0 ) -2 - (-3) + -3 - 2 ___ √( 1 ) + ( -5 ) = √ 2 2 ―― 26 ―――――――――――― 2 2 √( ) (0 2 - (-2) + ___ √( 4 ) + ( 4 ) = √ 2 2 ) - (-4) ―― 32 _ _ EF and GH do not have the same length. EF ≠ GH So, . Therefore, Module1 GE_MNLESE385795_U1M01L1.indd 9 9 Lesson1 3/20/14 5:02 PM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A DO NOT Correcti Reflect 7. Consider how the Distance Formula is related to the Pythagorean 4 Theorem. To use the Distance Formula to____ find the distance from 2 2 U(−3, −1) to V(3, 4), you write UV = √(3 - (-3)) + (4 - (-1)) . Explain how (3 - (−3)) in the Distance Formula is related to a in the Pythagorean Theorem and how (4 - (−1)) in the Distance Formula is y V x -4 U related to b in the Pythagorean Theorem. 0 4 -4 The Pythagorean Theorem states that a 2 + b 2 = c 2, where a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse. Applying this to the right ――― triangle in the figure, UV = c = √a 2 + b 2 , where a is the length of the horizontal leg of the triangle, or (3 - (−3)), and b is the length of the vertical leg of the triangle, or (4 - (−1)). Your Turn 8. _ _ Determine whether JK and LM have the same length. Justify your answer. J(−4, 4), K(−2, 1), L(−1, −4), M(2, −2) ―――――――― 2 ― 2 JK = (-2 - (-4)) + (1 - 4) = √13 ―――――――――― 2 2 ― LM = (2 - (-1)) + (-2 - (-4)) = √13 ― ― ― So, JK = LM = √13 . Therefore, JK and LM have the same length. √ √ 4 R K -4 0 P y S x Q M 4 L © Houghton Mifflin Harcourt Publishing Company Explain 3 J Finding a Midpoint The midpoint of a line segment is the point that divides the segment into two segments that have the same length. A line, ray, or other figure that passes through the midpoint of a segment is a segment bisector . _ In the figure, _ the tick marks show that PM = MQ. Therefore, M is the midpoint of PQ and line ℓ bisects PQ. ℓ P M Q You can use paper folding as a method to construct a bisector of a given segment and locate the midpoint of the segment. Module 1 GE_MNLESE385795_U1M01L1.indd 10 10 Lesson 1 3/20/14 5:02 PM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A Example 3 Use paper folding to construct a bisector of each segment. B B B A A A Step 1 Use a compass and straightedge _ to copy AB on a piece of paper. Fold the paper so that point B is on top of point A. Step 2 B AB A Step 3 A Open the paper. Label the point where the crease intersects the segment as point M. B B A M M A A _ _ Point M is the midpoint of AB and the crease is a bisector of AB. © Houghton Mifflin Harcourt Publishing Company Step 1 Step 2 Step 3 B _ M straightedge to copy JK on a piece of paper. Use a compass and A Fold the paper so that point K is on top of point . Open the paper. Label the point where the crease intersects the segment as point N. Point N is the midpoint Step 4 J J ― of JK and the crease is a bisector N K ― of JK. Make a sketch of your paper folding construction or attach your folded piece of paper. Reflect 9. Explain how you could use paper folding to divide a line segment into four segments of equal length. Use paper folding to construct the midpoint of the segment. Then use the same methods to construct the midpoint of each of the two new segments. The three midpoints divide the given segment into four segments of equal length. Module1 GE_MNLESE385795_U1M01L1 11 11 Lesson1 23/03/14 4:56 AM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A DO NOT Correcti Your Turn 10. Explain how to use a ruler to check your construction in Part B. Measure each of the segments formed by the bisector. The two segments should each have a length that is half as long as the given segment. Finding Midpoints on the Coordinate Plane Explain 4 You can use the Midpoint Formula to find the midpoint of a segment on the coordinate plane. The Midpoint Formula _ The midpoint M of AB with endpoints A(x 1, y 1)and B(x 2, y 2) y1 + y2 x1 + x2 _ , is given by M _ . 2 2 ( y ) B(x2, y2) M A(x1, y1) ( x +2 x , y +2 y ) 1 2 1 2 x Example 4 Show that each statement is true. _ _ PQ has endpoints P(-4, 1)and Q(2, −3) , then the midpointM of PQ lies in Quadrant III. If Use the Midpoint Formula to find the midpoint of ( ― ) -4 + 2 1 + (-3) M _, _ = M(-1, -1) 2 2 PQ. Substitute the coordinates, then simplify. So M lies in Quadrant III, since the x- and y-coordinates are both negative. _ _ RS has endpoints R(3, 5)and S(−3, −1) , then the midpointM of RS lies on the y-axis. If ( ) (0 5 + -1 3 + -3 _ , M _ =M 2 2 _ RS. Substitute the coordinates, then simplify. , 2 ) So M lies on the y-axis, since the x-coordinate is 0 . Your Turn Show that each statement is true. _ 11. If AB has endpoints A(6,_ −3)and B(-6, 3), then the midpoint M of AB is the origin. _ 12. If JK has endpoints J(7, 0)and K(−5, −4), _ then the midpoint M of JK lies in Quadrant IV. 6 + (-6) _ -3 + 3 M _ = M(0, 0) , 2 2 7 + (-5) 0 + (-4) M _, _ = M(1, -2) 2 2 ( ) ( So M is the origin, since the xand y-coordinates are both 0. Module1 GE_MNLESE385795_U1M01L1 12 ) © Houghton Mifflin Harcourt Publishing Company Use the Midpoint Formula to find the midpoint of So M lies in Quadrant IV, since the x-coordinate is positive and the y-coordinate is negative. 12 Lesson1 23/03/14 4:57 AM

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