1) 4(7 + 4) = 2(x + 6) 2) x(x + 5) = (x + 1)(x + 2) 3) (x + 2)(x + 5) = 2x(x + 2) 10.6 Warmup Solve the equation. Tuesday, March 24, 2:56

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1 10.6 Warmup Solve the equation. 1) 4(7 + 4) = ( + 6) ) ( + 5) = ( + 1)( + ) 3) ( + )( + 5) = ( + ) 1

2 Geometry 10.6 Segment Relationships in Circles

3 10.6 Essential Question What relationships eist among two intersecting chords or among segments that intersect outside a circle? 3

4 Goals Find the lengths of segments of chords. Find the lengths of segments and tangents. -Day Lesson 4

5 Quadratic Equation Review A quadratic equation is in the form a b c 0 Any quadratic equation can be solved using the formula b b 4ac a 5

6 Solve: 15 0 b b 4ac a (1) 4(1)( 15)

7 Solve: 15 0 b b 4ac a (1) 4(1)( 15)

8 Segments of a Chord Theorem (Chord-Chord) If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. 8

9 Chord Chord (simplified) a c d b a b = c d 9

10 Eample 1 Find a a = 8 a 40 = 8a 5 = a 10

11 Your Turn: Find. A E 6 4 B D 3 = = 48 = 16 = 4 C Check: 1 4 = 48 and 8 6 = 48 11

12 Eample Find ML and JK. 1

13 Proof R 3 S O 1 4 V T 3 and 4 both intercept arc SV. What does this tell use about 3 and 4? They are congruent. What kind of angles are 1 and? Vertical Angles And vertical angles are. Congruent. 13

14 Proof continued R Now SOR ~ VOT. Why? AA~ Postulate. 3 O 1 4 T In similar triangles, sides are proportional: OR OT = OS OV S V OR OV = OT OS 14

15 Terminology This line is a secant. This segment is a secant segment. 15

16 Terminology This segment is the eternal secant segment. 16

17 Terminology This segment is a tangent segment. This line is a tangent. 17

18 Terminology AC is a. secant segment AB is the. eternal secant segment AD is a. tangent segment C B A D 18

19 Tangent Secant Theorem AD AB AC C B A D 19

20 Tangent Secant (simplified) c = a(a + b) b a c 0

21 Eample 3 Find AD. AD 4(4 6) 4(10) C 40 6 AD B AD 10 A D 1

22 Your Turn. Solve for. 4(8)

23 Turn it up a notch Solve for. 5 ( 4) 5 4 Now What? 5 4 3

24 Quadratic Equation 5 ( 4) 5 4 Set quadratic equations equal to zero

25 Quadratic Formula b b 4ac a a = 1 b = 4 c = -5 5

26 Quadratic Formula ( 5) a = 1 b = 4 c = -5 6

27 Solve it. = b ± b 4ac a (1) 4 4 4(1)( 5) can t be negative

28 Just do it! 8

29 Your Turn Solve for. 3 Equation: 3 = ( + ) + 9

30 3 Solution 3 = ( + ) 9 = + 0 = + 9 a = 1 b = c = -9 (1) 4(1)( 9)

31 Secant Secant Theorem a b c d a(a+b) = c(c+d) 31

32 Eample 4 Solve for. Solution: 5(5 + 8) = 6(6 + ) 5(13) = = = 6 = 9 6 (or 4.83) X 3

33 Your Turn Solve for X 33

34 Solution X 9(9 11) 10(10 ) 9(0)

35 Eample 5 Solve for Equation: 5 = 4(16) Why? 5 = 64 5 = 1.8 X 35

36 Summary Segments in a circle have three situations: Chord Chord Tangent Secant Secant Secant Do you know the formula for each? Read the problems carefully. Use the correct numbers for each variable. 36

37 Formula Summary Chord Chord Tangent - Secant Secant Secant b a d b a c b a c d c ab = cd a(a+b) = c(c+d) c = a(a + b) 37

38 Homework 38

4.2 Rolle's Theorem and Mean Value Theorem

4.2 Rolle's Theorem and Mean Value Theorem Rolle's Theorem: Let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). If f (a) = f (b), then there is at least one