10.7 Special Segments in a Circle
Glencoe Geometry Interactive Chalkboard
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Columbus, Ohio 43240
Objectives
• Find measures of segments that intersect
in the interior of a circle.
• Find measures of segments that intersect
in the exterior of a circle.
Segments in a Circle
Theorem 10.15:
If two chords intersect in a circle, then the
products of the measures of the segments of
the chords are equal.
A
AO • OB = CO • OD
C
D
O
B
Find x.
Theorem 10.15
Multiply.
Divide each side by 8.
Answer: 13.5
Find x.
Answer: 12.5
BIOLOGY Biologists often examine organisms under
microscopes. The circle represents the field of view
under the microscope with a diameter of 2 mm.
Determine the length of the organism if it is located
0.25 mm from the bottom of the field of view. Round to
the nearest hundredth.
Draw a model using a circle. Let x represent the unknown
measure of the equal lengths of the chord
which is
the length of the organism. Use the products of the
lengths of the intersecting chords to find the length of the
organism. Note that
Segment products
Substitution
Simplify.
Take the square root of each
side.
Answer: 0.66 mm
ARCHITECTURE Phil is installing a new window in an
addition for a client’s home. The window is a rectangle
with an arched top called an eyebrow. The diagram
below shows the dimensions of the window. What is
the radius of the circle containing the arc if the
eyebrow portion of the window is not a semicircle?
Answer: 10 ft
Segments Outside of a Circle
Theorem 10.16:
If two secants intersect outside a circle, then the
product of the measures of the external secant
segment and the entire secant segment is equal to
the product of the measures of the other external
secant segment and its secant segment.
Z
OW• OZ = OY • OX
W
O
Y
X
Find x if EF 10, EH 8, and FG 24.
Secant Segment Products
Substitution
Distributive Property
Subtract 64 from each side.
Divide each side by 8.
Answer: 34.5
Find x if
Answer: 26
and
Segments Outside of a Circle
Theorem 10.17:
If a tangent segment and a secant segment
intersect outside a circle, then the square of the
measure of the tangent segment is equal to the
product of the measures of the secant segment
and its external segment.
Z
OZ• OZ = OX • OY
O
Y
X
Find x. Assume that segments that appear to be
tangent are tangent.
Disregard the negative solution.
Answer: 8
Find x. Assume that segments that appear to be
tangent are tangent.
Answer: 30
Assignment
• Pre-AP Geometry
Pg. 572 #8 - 29
• Geometry:
Pg. 572 #8 – 19, 22 - 28
10.1 Circles and Circumference
Glencoe Geometry Interactive Chalkboard
Copyright © by The McGraw-Hill Companies, Inc.
Developed by FSCreations, Inc., Cincinnati, Ohio 45202
Send all inquiries to:
GLENCOE DIVISION
Glencoe/McGraw-Hill
8787 Orion Place
Columbus, Ohio 43240
Objectives
• Recognize and apply properties of
trapezoids
• Solve problems using the medians of
trapezoids
Example 1 Equation of a Circle
Example 2 Use Characteristics of Circles
Example 3 Graph a Circle
Example 4 A Circle Through Three Points
Write an equation for a circle with the center at
(3, –3), d 12.
Equation of a circle
Simplify.
Answer:
Write an equation for a circle with the center at
(–12, –1), r 8.
Equation of a circle
Simplify.
Answer:
Write an equation for each circle.
a. center at (0, –5), d 18
Answer:
b. center at (7, 0), r 20
Answer:
A circle with a diameter of 10 has its center in the
first quadrant. The lines y –3 and x –1 are
tangent to the circle. Write an equation of the circle.
Sketch a drawing of the two
tangent lines.
Since d 10, r 5. The line x –1 is perpendicular to a
radius. Since x –1 is a vertical line, the radius lies on a
horizontal line. Count 5 units to the right from x –1.
Find the value of h.
Likewise, the radius perpendicular to the line y –3 lies
on a vertical line. The value of k is 5 units up from –3.
The center is at (4, 2),
and the radius is 5.
Answer: An equation for the circle is
.
A circle with a diameter of 8 has its center in the
second quadrant. The lines y –1 and x 1 are
tangent to the circle. Write an equation of the circle.
Answer:
Graph
Compare each expression in the equation to the standard
form.
The center is at (2, –3), and the radius is 2.
Graph the center. Use a compass set at a width
of 2 grid squares to draw the circle.
Answer:
Graph
Write the expression in standard form.
The center is at (3, 0), and the radius is 4.
Draw a circle with radius 4, centered at (3, 0).
Answer:
a. Graph
Answer:
b. Graph
Answer:
ELECTRICITY Strategically located substations are
extremely important in the transmission and
distribution of a power company’s electric supply.
Suppose three substations are modeled by the
points D(3, 6), E(–1, 0), and F(3, –4). Determine the
location of a town equidistant from all three
substations, and write an equation for the circle.
Explore You are given three points that lie on a circle.
Plan
Graph DEF. Construct the perpendicular
bisectors of two sides to locate the center, which
is the location of the tower. Find the length of a
radius. Use the center and radius to write an
equation.
Solve
Graph DEF and construct the perpendicular
bisectors of two sides. The center appears to be
at (4, 1). This is the location of the tower. Find r
by using the Distance Formula with the center
and any of the three points.
Write an equation.
Examine You can verify the location of the center by
finding the equations of the two bisectors and
solving a system of equations. You can verify
the radius by finding the distance between the
center and another of the three points on the
circle.
Answer:
AMUSEMENT PARKS The designer of an
amusement park wants to place a food court
equidistant from the roller coaster located at (4, 1),
the Ferris wheel located at (0, 1), and the boat ride
located at (4, –3). Determine the location for the
food court and write an equation for the circle.
Answer:
Assignment
• Pre-AP Geometry
Pg.
• Geometry:
Pg.