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9 th Grade Geometry Lesson 10-5: Tangents

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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

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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

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Example: Find Lengths ALGEBRA RS is tangent to Q at point R. Find y. ALGEBRA RS is tangent to Q at point R. Find y. y 20 16 S RP Q Because the radius is perpendicular to the tangent at the point of tangency, QR SR. 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.

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Example: Find Lengths (SR) 2 (SR) 2 + (QR) 2 = (SQ) 2 Pythagorean Theorem 16 + (QR) = 20 16 2 + (QR) 2 = 20 2 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

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

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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

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

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Example: Identify Tangents (AB) 2 + (BC) 2 = (AC) 2 Converse of the Pythagorean Theorem 7 2 + 9 2 = 14 2 AB = 7, BC = 9, AC = 14 7 2 + 9 2 = 14 2 AB = 7, BC = 9, AC = 14 130 ≠ 196Simplify 130 ≠ 196Simplify 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. ? ?

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

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Example: Identify Tangents (DW) 2 + (EW) 2 = (DE) 2 Converse of the Pythagorean Theorem 10 2 +24 2 = 26 2 DW = 10, EW = 24, DE = 26 10 2 +24 2 = 26 2 DW = 10, EW = 24, DE = 26 676 = 676Simplify. 676 = 676Simplify. 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. ? ?

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

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

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Theorem 10.11 If two segments from the same exterior point are tangent to a circle, then they are congruent –Example: AB ≈ AC B CA

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Example: Congruent Tangents ALGEBRA Find x. Assume that segments that appear tangent to circles are tangent. H G D 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 F E10 y x + 4 y - 5

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Example: Congruent Tangents ED = FDDefinition of congruent segments 10 = ySubstitution Use the value of y to find x. DG = DHDefinition of congruent segments DG = DHDefinition of congruent segments 10 + (y - 5) = y + (x + 4)Substitution 10 + (10 - 5) = 10 + (x + 4) y = 10 15 = 14 + x Simplify. 1 = xSubtract 14 from each side Answer: 1

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

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

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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 perimeter = 16 + 45 + 18 + 45 + 16 + 18 or 158 Substitution Answer: The perimeter of HJK is 158 units.

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

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