liedmanmath.weebly.com · Web viewCenter (-3, 5) Radius 6Center: (6, 0) Radius:3 (x + 3)² + (y-5)² = 36(x-6)² + y² = 9 Ex: What is the standard equation of the circle? (x-2)²

12.1 Tangent Lines Liedman

Tangent line: A line that intersects a circle once.

Point of tangency: The point where a circle and a tangent intersect. (T)

Ex:

90 + 90 + 152 + x = 360

332 + x = 360

X = 28

Ex: CB is tangent to Circle O. Find the value of x.

X = 180-90-26 = 64

Ex:

x² + 24² = (x+18)²

x² + 576 = x² + 36x + 324

252 = 36x

X = 7

Ex: What is the value of x of Circle D?

x² + 12² = (x+8)²x² + 144 = x² + 16x + 6480 = 16xX = 5

Ex:

5² + 12² = 8²169 ≠ 64MN is not tangent since it is not perpendicular.

Ex: Circle O is inscribed in ΔABC. Find the perimeter of the ΔABC.

10 + 10 + 15+ 15 + 8 + 8 = 66 cm

7² + 24² = 25²49 + 576 = 625625 = 625MN is tangent since it is perpendicular.

12.2 Chords and Arcs

Chord: A segment whose endpoints are on a circle

Ex: The circles are congruent. What can you conclude?

Ex: What is the length of RS? Ex: What is x?

25 x = 16

Ex: Find the missing variable.

3² + 7² = r² 11² + y ² = 15²

r = 7.6 y = 10.2

X = 1

12.3 Inscribed Angles

Inscribed angle: An angle whose vertex is on the circle and sides are chords.

Intercepted arc: An arc whose endpoints are the inscribed angle.

Ex: Find the missing angles.

50 a = 120 b= 75 m<A = 95, m<B = 77, m<C = 85, m<D = 103

Ex: Find the missing variables.

5x+110=180 x= 14 a = 180-68 = 112

104=2y= 180 y=38 b = 224 – 104 = 120

c = 142 – 104 = 38

Ex: What is the m<A? 90º Ex: What is the measure of <2? 38º

43 x= 16 y = 14

Ex:

M<SQP = ½(212) = 106º

M<PQR = 74º

12.4 Angle Measures and Segment Lengths

Ex: Find the value of the variables.

X = ½(112+140) x = ½(90-20)

X = 126 x = 35

20 = ½(95-z) 35 = ½(x + 30)

40 = 95 – z 70 = x + 30

Z = 55 x = 40

X = ½(240-120) m<1 = ½(170-55)

X = 60 m<1 = 57.5

6.5m = 3(7) (20+14)14 = (x+16)16 (8+6)6 = (y+7)7 (16+8)8 = z²M = 3.2 x = 13.8 y = 5 z = 13.9

12.5 Circles in the Coordinate Plane

Ex: What is the standard equation of a circle.

Center (-3, 5) Radius 6 Center: (6, 0) Radius:3

(x + 3)² + (y-5)² = 36 (x-6)² + y² = 9

Ex: What is the standard equation of the circle?

(x-2)² + (y-2)² = 16 x² + (y-3)² = 16

Ex: Find the center and radius of each circle. Then graph the circle.

x² + (y-2)² = 49 (x-4)² + (y + 3)² = 16

Center: (0,2) Center: (4,-3)

Radius: 7 Radius: 4

Ex: Write the standard equation of the circle with the given center and passes through the given point.

Center: (-2,6) Point (-2, 10)

1.) Find the radius using the distance formula d=√(x1−x2)2+( y1− y2)

2

d = √ (−2— 2 )2+(6−10) ²=4

2. Write equation using radius and the center.

(x+ 2)² + (y – 6)² = 16

Center: (4,3) Point (-1,1)

1. Radius: √ (4−−1 )2+(3−1) ²=√29

2. (x-4)² + (y-3)² = 29