Angle Measurements and Segment Lengths

1
Section 11-4

• Angle Measurements and Segment Lengths

2
Secant

• A secant is a line that intersects a circle at
  two points.

3
Theorem 11-11

• The measure of an angle formed by two lines that
• 1) Intersect inside a circle is half the sum of
  the measures of the intercepted arcs.

mlt1 ½( x y )
1
y
x
4
Theorem 11-11 continued

• 2)Intersect outside a circle is half the
  difference of the measures of the intercepted
  arcs.

1
y
m / 1½( x y )
1
1
y
x
x
5
Apply

• You focus your camera on a statue. Your camera
  is the vertex of the angle formed by the tangents
  to the statue. You estimate that this angle is
  50. What is the measure of the arc of the
  circular base of the statue that will be in your
  photo?

50 ½((360-x)-x) 50½(360-2x) 50180-x -130-x 130
x
50
x
6
Apply theorem 11-11

• Find x in each picture

x
80
160
40
x
105
40
x½( 160 80) 120
x½(105-40)32.5
7
Find the measure of an arc
20 ½ (95 z)20 47.5 ½ z-27.5 .5z55 z
20º
zº
95º
8
Theorem 11-12

• For a given point and circle, the product of the
  lengths of the two segments from the point to the
  circle is constant along any line through the
  point and circle.

r
w
x
P
a
c
P
P
s
z
d
t
b
y
a d b c
(s t) s r² (r r)
(w x) x (y z ) z
9
Apply Theorem 11-12

• Find x

5
7
8
4
10
x
6
x
(7 5) 5 (x 4) 4 60 4x 16 44 4x 11x
10 6 8x 60 8x 15/2 x
10
Challenge Problem
6

• Find x

(9 x) x 6 6 9x x² 36 x² 9x 36 0 (x
12) (x 3) 0 x 12 0 x-3 0 x -12,
x 3 Dont want negative answer x 3
9
x
11
Homework

• Page 611
• Problems 1,2,5,6,10,11,13-16,18,19,21,22,24,25,2
  7