DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A Name Class Date 15.3 Tangents and Circumscribed Angles Essential Question: What are the key theorems about tangents to a circle? Explore Resource Locker Investigating the Tangent-Radius Theorem A tangent is a line in the same plane as a circle that intersects the circle in exactly one point. The point where a tangent and a circle intersect is the point of tangency. P In the figure, the line is tangent to circle C, and point P is the point of tangency. You can use a compass and straightedge to construct a circle and a line tangent to it. Use a compass to draw a circle. Label the center C. B Mark a point P on the circle. Using a straightedge, draw a tangent to the circle through point P. Mark a point Q at a different position on the tangent line. D © Houghton Mifflin Harcourt Publishing Company Q A C C P C _ Use a straightedge to draw the radius CP. Use a protractor to measure ∠CPQ. Record the result in the table. Repeat the process two more times. Make sure to vary the size of the circle and the location of the point of tangency. Vehicle 1 Measure of ∠CPQ Circle 1 Circle 2 Circle 3 90° 90° 90° Reflect 1. Make a Conjecture Examine the values in the table. Make a conjecture about the relationship between a tangent line and the radius to the point of tangency. A tangent line and the radius to the point of tangency are perpendicular to each other. 2. Discussion Describe any possible inaccuracies related to the tools you used in this Explore. It is difficult to accurately construct the tangent using just a straightedge and a point on the circle. This may cause the angle measurements to vary slightly from 90°. Module 15 GE_MNLESE385801_U6M15L3.indd 805 805 Lesson 3 02/04/14 11:44 AM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A Explain 1 DO NOT Correcti Proving the Tangent-Radius Theorem The Explore illustrates the Tangent-Radius Theorem. Tangent-Radius Theorem If a line is tangent to a circle, then it is perpendicular to a radius drawn to the point of tangency. m Example 1 Complete the proof of the Tangent–Radius Theorem. Given: Line m is tangent to circle C at point P. P ¯⊥ m Prove: CP C ¯ is not perpendicular to line m. Use an indirect proof. Assume that CP ¯ ⊥ m. There must be a point Q on line m such that CQ ¯ ⊥ m, then △CQP is a right triangle, and CP > CQ because CQ ¯ hypotenuse of the right triangle. CP is the If C m are in the exterior of the circle. This means point Q is in the exterior of the circle. You can conclude that CP < CQ because ¯ CP is a radius of circle C. © Houghton Mifflin Harcourt Publishing Company P Since line m is a tangent line, it can intersect circle C at only point P , and all other points of line m Q This _ contradicts the initial assumption that a point Q exists such that CQ ⊥ m, because that meant that CP > CQ. Therefore, the assumption ¯ is false and CP must be perpendicular to line m. Reflect 3. _ Both lines in the figure are tangent to the circle, and AB is a diameter. What can you conclude about the tangent lines? Since the tangent lines are both perpendicular to ¯ AB, they are A parallel to each other. B Module15 GE_MNLESE385801_U6M15L3.indd 806 806 Lesson3 3/24/14 11:06 PM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A The converse of the Tangent-Radius Theorem is also true. You will be asked to prove this theorem as an exercise. Converse of the Tangent-Radius Theorem If a line is perpendicular to a radius of a circle at a point on the circle, then it is tangent to the circle at that point on the circle. Explain 2 Constructing Tangents to a Circle From a point outside a circle, two tangent lines can be drawn to the circle. Example 2 Use the steps to construct two tangent lines from a point outside a circle. Use a compass to draw a circle. Label the center C. A C M X B Mark a point X outside _the circle and use a straightedge to draw CX. Use a compass and_straightedge to construct the midpoint of CX and label the midpoint M. Use a compass to construct a circle with center M and radius CM. of circle C and circle M as A and B. Label the points of intersection ‹ › ‹ › − − © Houghton Mifflin Harcourt Publishing Company Use a straightedge to draw XA and XB . Both lines are tangent to circle C. Reflect 4. 5. _ ‹ › ‹ › − − How can you justify that XA (or CA on the diagram.) _XB ) is a tangent line? (Hint: Draw _ The endpoints of diameter CX lie on the inscribed angle ∠CAX of circle M. By the _ Angle Inscribed in a Semicircle Theorem, ∠CAX is a right angle. By the Converse of the ‹ › − Tangent-RadiusTheorem, XA is a tangent line to circle C. The same reasoning shows ‹ › − that XB is a tangent line. _ _ Draw CA and CB on the diagram. Consider quadrilateral CAXB. State any conclusions you can reach about the measures of the angles of CAXB. The sum of the measures of the angles is 360°. There are two right angles, ∠CAX and ∠CBX. ∠AXB and ∠ACB are supplementary. Module15 GE_MNLESE385801_U6M15L3.indd 807 807 Lesson3 23/03/14 2:00 AM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A Explain 3 DO NOT Correcti Proving the Circumscribed Angle Theorem A circumscribed angle is an angle formed by two rays from a common endpoint that are tangent to a circle. Circumscribed Angle Theorem A circumscribed angle of a circle and its associated central angle are supplementary. Example 3 Prove the Circumscribed Angle Theorem. Given: ∠AXB is a circumscribed angle of circle C. A Prove: ∠AXB and ∠ACB are supplementary. _ _ Since ∠AXB is a circumscribed angle of circle C, XA and XB are C X tangents to the circle. Therefore, ∠XAC and ∠XBC are B right angles by the Tangent-Radius Theorem . In quadrilateral XACB, the sum of the measures of its four angles is 360° . Since m∠XAC + m∠XBC = 180° , this means m∠AXB + m∠ACB = 360° - 180° = 180° . So, ∠AXB and ∠ACB are supplementary by the definition of supplementary angles . Reflect 6. Is it possible for quadrilateral AXBC to be a parallelogram? If so, what type of parallelogram must it be? If not, why not? Yes. If AXBC is a parallelogram, it must be a square. → → ‾_and KN ‾ are tangent to circle C. Explain how to show that 1. KM _ KM ≅ _ KN, using congruent triangles. Draw KC. This creates two right triangles, △KCM and _ _ _ △KCN.Since ¯ CM ≅ CN andCK ≅ CK, the triangles are congruent by the Hypotenuse-Leg Theorem. _ It follows that ¯ KM ≅ KN by CPCTC. 2. M C K N Essential Question Check-In What are the key theorems regarding tangent lines to a circle? If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of © Houghton Mifflin Harcourt Publishing Company Elaborate tangency. Conversely, if a line is perpendicular to a radius of a circle at a point on the circle, it is tangent to the circle. Also, a circumscribed angle of a circle and its associated central angle are supplementary. Module15 GE_MNLESE385801_U6M15L3.indd 808 808 Lesson3 23/03/14 1:59 AM

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