Sec. 12 – 4 Measures & Segment Lengths in Circles

Secants

EA

B

F

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

•EF or AB are secants

•AB is a chord

Thm 11 – 11: The measure of an formed by 2 lines that intersect inside a circle is

m1 = ½(x + y)

Measure of intercepted arcs1 x°

y°

(…Thm 11 – 11 Continues) The measure of an formed by 2 lines that intersect outside a circle is m1 = ½(x - y)

Smaller ArcLarger Arc

x°

y°

1

x°

y°

1

2 Secants:

x°

y°

1

Tangent & a Secant

2 Tangents

3 cases:

Ex.1 & 2:

• Find the measure of arc x.

• Find the mx.

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°

Thm (11 – 12) Lengths of Secants, Tangents, & Chords

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 & 4• Find length of x.

• 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 Secants

• 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!• Find 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°

Homework: p. 691 #1-6, 9-14, 21, 25