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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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## 9 th Grade Geometry

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**9th 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**• 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**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.**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**Example**CD is a tangent to B at point D. Find a. • 15 • 20 • 10 • 5 C a B A D 40 25**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**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**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. ? ?**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**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. ? ?**Quick Review**Determine whether ED is a tangent to Q. A. Yes B. No C. Cannot be determined D √549 18 Q E 15**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**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**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**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**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**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**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.**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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