Angle Measures and Angle Measures and Segment Lengths in CirclesSegment Lengths in Circles

Objectives:Objectives:1) To find the measures of 1) To find the measures of s formed s formed

by chords, secants, & tangents.by chords, secants, & tangents.2) To find the lengths of segments 2) To find the lengths of segments

associated with circles.associated with circles.

SecantsSecants

EA

B

F

Secant – A line that intersects a circle in exactly 2 points.

•EF or AB are secants

•AB is a chord

Theorem. The measure of an Theorem. The measure of an formed formed by 2 lines that intersect by 2 lines that intersect insideinside a circle isa circle is

m1 = ½(x + y)

Measure of intercepted arcs1 x°

y°

Theorem. The measure of an Theorem. The measure of an formed formed by 2 lines that intersect by 2 lines that intersect outsideoutside a circle isa circle is

m1 = ½(x - y) Smaller Arc

Larger Arc

x°

y°

1

x°

y°

1

2 Secants:

x°

y°

1

Tangent & a Secant

2 Tangents

3 cases:

Ex.1 & 2:Ex.1 & 2: Find the measure of Find the measure of

arc x.arc x.

Find the mFind the mx.x.

94°

112°

x°

m1 = ½(x + y)

94 = ½(112 + x)

188 = (112 + x)

76° = x

68° 104°

92°

268°

x°

mx = ½(x - y)

mx = ½(268 - 92)

mx = ½(176)

mx = 88°

Lengths of Secants, Tangents, & Lengths of Secants, Tangents, & ChordsChords

2 Chords

a c

b

d

a•b = c•d

2 Secants

x

w

z

y

w(w + x) = y(y + z)

Tangent & Secant

t

y

z

t2 = y(y + z)

Ex. 3 & 4Ex. 3 & 4 Find length of x.Find length of x.

Find the length of g.Find the length of g.

3 x

7

5

a•b = c•d

(3)•(7) = (x)•(5)

21 = 5x

4.2 = x

15

8

g

t2 = y(y + z)

152 = 8(8 + g)

225 = 64 + 8g

161 = 8g

20.125 = g

Ex.5: 2 SecantsEx.5: 2 Secants

Find the length of x.Find the length of x.

14

20

16

x

w(w + x) = y(y + z)

14(14 + 20) = 16(16 + x)

(34)(14) = 256 + 16x

476 = 256 + 16x

220 = 16x

3.75 = x

Ex.6: A little bit of everything!Ex.6: A little bit of everything!Find the measures of the missing variablesFind the measures of the missing variables

9

12

k

8

a°r

60°

175°

Solve for k first.

w(w + x) = y(y + z)

9(9 + 12) = 8(8 + k)

186 = 64 + 8k

k = 15.6

Next solve for r

t2 = y(y + z)

r2 = 8(8 + 15.6)

r2 = 189

r = 13.7

Lastly solve for ma

m1 = ½(x - y)

ma = ½(175 – 60)

ma = 57.5°

What have we learned??What have we learned??

When dealing with angle measures formed by When dealing with angle measures formed by intersecting secants or tangents you either add intersecting secants or tangents you either add or subtract the intercepted arcs depending on or subtract the intercepted arcs depending on where the lines intersect.where the lines intersect.

There are 3 formulas to solve for segments There are 3 formulas to solve for segments lengths inside of circles, it depends on which lengths inside of circles, it depends on which segments you are dealing with: Secants, segments you are dealing with: Secants, Chords, or Tangents.Chords, or Tangents.