10.5 Warmup. Find the indicated measure. Thursday, March 23, 2:46. Geometry 10.5 Angle Relationships in Circles 1

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1 0.5 Warmup Find the indicated measure.

2 0.5 Warmup Solve each equation. Use Factoring. ) x 2 + 4x + 40 = 0 2) x 2 36 = 0 3) x 2 + 5x = 0 4) x 2 + 5x = 24 5) 3x x = 48 6) 5x 2 = 20 2

3 0.5 Warmup (Answers) Solve each equation. Use Factoring.. x 2 + 4x + 40 = 0 2. x 2 36 = 0 x = 4 x = 0 3. x 2 + 5x = 0 4. x 2 + 5x = 24 x = 0 x = 5 x = ±6 x = 3 x = x x = x 2 = 20 x = 4 x = ±4 3

4 Geometry 0.5 Angle Relationships in Circles

5 0.5 Essential Question When a chord intersects a tangent line or another chord, what relationships exist among the angles and arcs formed? 5

6 Goal o Use angles formed by tangents, secants, and chords to solve problems. 6

7 x Review Note: in solving an equation with fractions, one of the first things to do is always clear the fractions x x x x

8 20 4x 0 You do it. Solve: x x x x x

9 If two lines intersect a circle, where can the lines intersect each other? Inside the circle. On the circle. Outside the circle. 9

10 0

11 Review: Inscribed Angle Thm 40 a 80 The measure of an inscribed angle is equal to one-half the measure of the intercepted arc. m = 2 a What if one side of the angle is tangent to the circle?

12 Tangent-Chord Theorem (0.2) C 2 A B If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one-half the measure of the intercepted arc. and 2 2 m mab m2 mbca 2

13 Simplified Formula b a 2 m m2 2 2 a b 3

14 Example C B 80 2 mab mab A 80 mbca Find the mab and mbca. 4

15 Example 2. Solve for x. C B 4 x (0 x60) 2 (0x 60) 8x0x60 2x 60 4x x 30 A 5

16 Chord-Chord Theorem (0.3) (Inside the circle) B A C If two chords intersect in a circle, then the measure of the angle is one-half the sum of the intercepted arcs. D 2 m mab mcd 6

17 Simplified Formula b a m a b 2 7

18 Example 3 Find m. 30 B A C 80 2 m (0) D 55 8

19 Example 4 Solve for x. 20 B A 60 C 00 x 60 x x x Check: D = = 60 9

20 Your turn. Solve for x & y. A C 20 P K 75 B x x x D y y 20 y O M y 20

21 2

22 Secant-Secant C D A B 2 m mab mcd 22

23 Simplified Formula b a 2 m a b 23

24 Secant-Tangent A C B 2 m mab mbc 24

25 Simplified Formula a b 2 m a b 25

26 Tangent-Tangent A C B 2 m macb mab 26

27 Simplified Formula b a m a b 2 27

28 Intersection Outside the Circle Secant-Secant Secant-Tangent Tangent-Tangent In all cases, the measure of the exterior angle is found the same way: One-half the difference of the larger and smaller arcs. 28

29 Example 5 Find m m 80 0 (70) 35 29

30 Example 6 Find m m (50) 25 30

31 Example 7 Rays k and m are tangent to the circle. 30 Find m. k? = m (60) 30 m 3

32 Example 8 The northern lights are bright flashes of colored light between 50 and 200 miles above Earth. A flash occurs 50 miles above Earth at point C. What is the measure of, the portion of Earth from which the flash is visible? (Earth s radius is approximately 4000 miles.) 32

Example 8 Answer m DAC cos x = 4000 450 x = cos

33 Example 8 Answer m DAC cos x = x = cos m DAB = 2m DAC m DAB x

34 How to remember this: o If the angle vertex is on the circle, its measure is one-half the intercepted arc. o If an angle vertex is inside the circle, its measure if one-half the sum of the intercepted arcs. o If an angle vertex is outside the circle, its measure is one-half the difference of the intercepted arcs. 34

35 What can you do? o Do problems 9. o Carefully consider what the situation is and use the correct formula. o 8 minutes. 35

36 Extra Problems Line m is tangent to the circle. Find the indicated measure.. m 2. 36

37 Extra Problems Find the value of the variable

38 Extra Problems Find the value of the variable

39 Extra Problems Find the value of the variable

40 Extra Problems 9. You are on top of Mount Rainier on a clear day. You are about 2.73 miles above sea level at point B. Find which represents the part of Earth that you can see. (Use Trig) 40

41 Answers. 05 [x=½ 20] [x=2 98] [x=½( )] 4. 6 [78=½(x + 95)] 5. 5 [x=½(78 76)] [30= ½(a 44)] [x= ½(225 35)] [30= ½((360-2x) 2x)] 9. 4 [2(cos x = )] 4

42 Homework 42

6.1 Circles and Related Segments and Angles

Chapter 6 Circles 6.1 Circles and Related Segments and Angles Definitions 32. A circle is the set of all points in a plane that are a fixed distance from a given point known as the center of the circle.