Transcript Angle Measures and Segment Lengths LESSON 12-4 Additional Examples Find the value of the variable. a. 2 HELP x = (268 – 92) The measure of an angle.

Angle Measures and Segment Lengths
LESSON 12-4
Additional Examples
Find the value of the variable.
a.
1
2
HELP
x = (268 – 92)
The measure of an angle formed by two
lines that intersect outside a circle is half
the difference of the measures of the
intercepted arcs (Theorem 12-11 (2)).
x = 88
Simplify.
GEOMETRY
Angle Measures and Segment Lengths
LESSON 12-4
Additional Examples
(continued)
b.
1
94 = 2 (x + 112)
1
94 = 2 x + 56
1
2
HELP
The measure of an angle formed by
two lines that intersect inside a circle
is half the sum of the measures of the
intercepted arcs (Theorem 12-11 (1)).
Distributive Property
38 = x
Subtract.
76 = x
Multiply each side by 2.
Quick Check
GEOMETRY
Angle Measures and Segment Lengths
LESSON 12-4
Additional Examples
An advertising agency wants a frontal photo of a “flying
saucer” ride at an amusement park. The photographer stands at the
vertex of the angle formed by tangents to the “flying saucer.” What is
the measure of the arc that will be in the photograph?
In the diagram, the photographer stands at point T.
TX and TY intercept minor arc XY and major arc XAY.
HELP
GEOMETRY
Angle Measures and Segment Lengths
LESSON 12-4
Additional Examples
(continued)
Let mXY = x.
Then mXAY = 360 – x.
m
T = 1 (mXAY – mXY)
2
1
2
1
72 = (360 – 2x)
2
72 = [(360 – x) – x]
The measure of an angle formed by two
lines that intersect outside a circle is half
the difference of the measures of the
intercepted arcs (Theorem 12-11 (2)).
Substitute.
Simplify.
72 = 180 – x
Distributive Property
x + 72 = 180
Solve for x.
x = 108
A 108° arc will be in the advertising agency’s photo.
HELP
Quick Check
GEOMETRY
Angle Measures and Segment Lengths
LESSON 12-4
Additional Examples
Find the value of the variable.
a.
5•x=3•7
5x = 21
Along a line, the product of the lengths
of two segments from a point to a circle
is constant (Theorem 12-12 (1)).
Solve for x.
x = 4.2
b.
8(y + 8) = 152
Along a line, the product of the lengths
of two segments from a point to a circle is
constant (Theorem 12-12 (3)).
8y + 64 = 225
Solve for y.
8y = 161
y = 20.125
HELP
Quick Check
GEOMETRY
Angle Measures and Segment Lengths
LESSON 12-4
Additional Examples
Quick Check
A tram travels from point A to point B along the arc of a circle
with a radius of 125 ft. Find the shortest distance from point A to point B.
The perpendicular bisector of the chord AB
contains the center of the circle.
Because the radius is 125 ft, the diameter is 2 • 125 = 250 ft.
The length of the other segment along the diameter is 250 ft – 50 ft, or 200 ft.
x • x = 50 • 200
x2 = 10,000
Along a line, the product of the lengths
of the two segments from a point to a
circle is constant (Theorem 12-12 (1)).
Solve for x.
x = 100
The shortest distance from point A to point B is 200 ft.
HELP
GEOMETRY