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9 th Grade Geometry. Lesson 10-5: Tangents. Main Idea . Use properties of tangents! Solve problems involving circumscribed polygons. New Vocabulary. Tangent Any line that touches a curve in exactly one place Point of Tangency The point where the curve and the line meet. Theorem 10.9.

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1. 9th Grade Geometry Lesson 10-5: Tangents

2. Main Idea • Use properties of tangents! • Solve problems involving circumscribed polygons New Vocabulary • Tangent • Any line that touches a curve in exactly one place • Point of Tangency • The point where the curve and the line meet

3. Theorem 10.9 • If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. • Example: If RT is a tangent, OR RT T R O

4. Example: Find Lengths ALGEBRARS is tangent to Q at point R. Find y. S 20 16 Q P R y Because the radius is perpendicular to the tangent at the point of tangency, QRSR. This makes SRQ a right angle and SRQ a right triangle. Use the Pythagorean Theorem to find QR, which is one-half the length y.

5. Example: Find Lengths (SR)2 + (QR)2= (SQ)2Pythagorean Theorem 162 + (QR)2 = 202 SR = 16, SQ = 20 256 + (QR)2 = 400 Simplify (QR)2 = 144 Subtract 256 from each side QR = +12 Take the square root of each side Because y is the length of the diameter, ignore the negative result. Thus, y is twice QR or y = 2(12) = 24 Answer:y = 24

6. Example CD is a tangent to B at point D. Find a. • 15 • 20 • 10 • 5 C a B A D 40 25

7. Theorem 10.10 • If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle. • Example: If OR RT, RT is a tangent. R T O

8. Example: Identify Tangents Determine whether BC is tangent to A C 7 9 7 A B 7 First determine whether ABC is a right triangle by using the converse of the Pythagorean Theorem

9. Example: Identify Tangents (AB)2 + (BC)2 = (AC)2 Converse of the Pythagorean Theorem 72 + 92 = 142AB = 7, BC = 9, AC = 14 130 ≠ 196 Simplify Because the converse of the Pythagorean Theorem did not prove true in this case, ABC is not a right triangle Answer:So, BC is not a tangent to A. ? ?

10. Example: Identify Tangents Determine whether WE is tangent to D. E 16 24 10 D W 10 First Determine whether EWD is a right triangle by using the converse of the Pythagorean Theorem

11. Example: Identify Tangents (DW)2 + (EW)2 = (DE)2 Converse of the Pythagorean Theorem 102 +242 = 262DW = 10, EW = 24, DE = 26 676 = 676 Simplify. Because the converse of the Pythagorean Theorem is true, EWD is a right triangle and EWD is a right angle. Answer:Thus, DW WE, making WE a tangent to D. ? ?

12. Quick Review Determine whether ED is a tangent to Q. A. Yes B. No C. Cannot be determined D √549 18 Q E 15

13. Quick Review Determine whether XW is a tangent to V. A. Yes B. No C. Cannot be determined W 10 17 10 V X 10

14. Theorem 10.11 • If two segments from the same exterior point are tangent to a circle, then they are congruent • Example: AB ≈ AC B C A

15. Example: Congruent Tangents ALGEBRA Find x. Assume that segments that appear tangent to circles are tangent. ED and FD are drawn from the same exterior point and are tangent to S, so ED ≈ FD. DG and DH are drawn from the same exterior point and are tangent to T, so DG ≈ DH H x + 4 F y D G y - 5 E 10

16. Example: Congruent Tangents ED = FD Definition of congruent segments 10 = y Substitution Use the value of y to find x. DG = DH Definition of congruent segments 10 + (y - 5) = y + (x + 4) Substitution 10 + (10 - 5) = 10 + (x + 4) y = 10 15 = 14 + x Simplify. 1 = x Subtract 14 from each side Answer:1

17. Quick Review Find a. Assume that segments that appear tangent to circles are tangent. • 6 • 4 • 30 • -6 30 N b 6 – 4a R A

18. Example: Triangles Circumscribed About a Circle Triangle HJK is circumscribed about G. Find the perimeter of HJK if NK = JL +29 H N 18 K L M 16 J

19. Example: Triangles Circumscribed About a Circle Use Theorem 10.11 to determine the equal measures: JM = JL = 16, JH = HN = 18, and NK = MK We are given that NK = JL + 29, so NK = 16 + 29 or 45 Then MK = 45 P = JM + MK + HN + NK + JL + LH Definition of perimeter = 16 + 45 + 18 + 45 + 16 + 18 or 158 Substitution Answer:The perimeter of HJK is 158 units.

20. Quick Review Triangle NOT is circumscribed about M. Find the Perimeter of NOT if CT = NC – 28. • 86 • 180 • 172 • 162 N 52 C T A B 10 O

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