Angle Measures and Segment Lengths in Circles

1
Angle Measures and Segment Lengths in Circles

• Objectives
• 1) To find the measures of ?s formed by
  chords, secants, tangents.
• 2) To find the lengths of segments
  associated with circles.

2
Secants
F
B
A
E

• Secant A line that intersects a circle in
  exactly 2 points.
• EF or AB are secants
• AB is a chord

3

• Theorem. The measure of an ? formed by 2 lines
  that intersect inside a circle is

m?1 ½(x y)
x
1
y
Measure of intercepted arcs
4

• Theorem. The measure of an ? formed by 2 lines
  that intersect outside a circle is

m?1 ½(x - y)
Smaller Arc
3 cases
Larger Arc
1
1
Tangent a Secant
2 Secants
y
1
y
y
x
2 Tangents
x
x
5
Ex.1 2

• Find the m?x.

• Find the measure of arc x.

x
x
92
104
68
94
268
112
m?x ½(x - y) m?x ½(268 - 92) m?x ½(176) m?x
88
m?1 ½(x y) 94 ½(112 x) 188 (112
x) 76 x
6
Lengths of Secants, Tangents, Chords
2 Secants
2 Chords
Tangent Secant
y
a
c
t
z
x
b
z
d
w
y
ab cd
t2 y(y z)
w(w x) y(y z)
7
Ex. 3 4

• Find the length of g.

• Find length of x.

8
15
g
3
x
7
5
t2 y(y z) 152 8(8 g) 225 64 8g 161
8g 20.125 g
ab cd (3)(7) (x)(5) 21 5x 4.2 x
8
Ex.5 2 Secants

• Find the length of x.

20
14
16
w(w x) y(y z) 14(14 20) 16(16
x) (34)(14) 256 16x 476 256 16x 220
16x 3.75 x
x
9
Ex.6 A little bit of everything!

• Find the measures of the missing variables

Solve for k first. w(w x) y(y z) 9(9 12)
8(8 k) 186 64 8k k 15.6
12
k
175
9
8
60
Next solve for r t2 y(y z) r2 8(8
15.6) r2 189 r 13.7
a
r
Lastly solve for m?a m?1 ½(x - y) m?a ½(175
60) m?a 57.5
10
What have we learned??

• When dealing with angle measures formed by
  intersecting secants or tangents you either add
  or subtract the intercepted arcs depending on
  where the lines intersect.
• There are 3 formulas to solve for segments
  lengths inside of circles, it depends on which
  segments you are dealing with Secants, Chords,
  or Tangents.