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Tangents and Circumscribed Polygons: Properties and Problem-solving

Learn about tangents, their properties, and how to solve problems involving circumscribed polygons. Explore the concept of a point of tangency and its significance. This lesson is part of Chapter 10-5 and aligns with Standard 7.0 and Standard 21.0.

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Tangents and Circumscribed Polygons: Properties and Problem-solving

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  1. Chapter 10-5 Tangents

  2. Use properties of tangents. • tangent • Solve problems involving circumscribed polygons. • point of tangency Standard 7.0Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles. (Key) Standard 21.0Students prove and solve problems regarding relationships among chords, secants, tangents,inscribed angles, and inscribed and circumscribed polygons of circles. (Key) Lesson 5 MI/Vocab

  3. Reminder • Tangent—a line that intersects the circle in only one point

  4. Secant Tangent Diameter Radius Point of Tangency Chord

  5. Tangent Theorem • A line is tangent to a circle  it is  to a radius at its endpoint on the circle

  6. Examples:

  7. Find Lengths Because y is the length of the diameter, ignore the negative result. Thus, y is twice QR or y = 2(12) = 24. Lesson 5 Ex1

  8. A • B • C • D A. 15 B. 20 C. 10 D. 5 Lesson 5 CYP1

  9. Identify Tangents Because the converse of the Pythagorean Theorem did not prove true in this case, ΔABC is not a right triangle. Lesson 5 Ex2

  10. Identify Tangents First determine whether ΔEWDis a right triangle by using the converse of the Pythagorean Theorem. Because the converse of the Pythagorean Theorem is true, ΔEWD is a right triangle and EWD is a right angle. Lesson 5 Ex2

  11. A • B • C A. yes B. no C. cannot be determined Lesson 5 CYP2

  12. A • B • C A. yes B. no C. cannot be determined Lesson 5 CYP2

  13. 12 r 10 r r2 + 122 = (r + 10)2 r2 + 144 = r2 + 20r + 100 144 = 20r + 100 44 = 20r r =

  14. C AC = AB A B • If two segments from the same external point are tangent to a circle  they are 

  15. Congruent Tangents ALGEBRA Find x.Assume that segments that appear tangent to circles are tangent. Lesson 5 Ex3

  16. 10 10 Congruent Tangents Use the value of y to find x. Answer: 1 Lesson 5 Ex3

  17. ALGEBRA Find a.Assume that segments that appear tangent to circles are tangent. • A • B • C • D A. 6 B. 4 C. 30 D. –6 Lesson 5 CYP3

  18. Triangles Circumscribed About a Circle Interactive Lab: Tangents and Communication Signals 16 45 16 + 29 = 45 18 P = 16 + 18 + 18 + 45 + 45 + 16 = 158 Lesson 5 Ex4

  19. A • B • C • D A. 86 B. 180 C. 172 D. 162 Lesson 5 CYP4

  20. Common External Tangent Common Internal Tangent

  21. Homework Ch 10-5 • Pg 593 5 – 22, 30, 31, 43 – 46

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