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Five-Minute Check (over Lesson 107) CCSS Then/Now
New Vocabulary Key Concept Equation of a
Circle in Standard Form Example 1 Write an
Equation Using the Center and Radius Example
2 Write an Equation Using the Center and a
Point Example 3 Graph a Circle Example
4 Real-World Example Use Three Points to Write
an Equation Example 5 Intersections with Circles
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5-Minute Check 1
Find x.
A. 1 B. 2 C. 3 D. 4
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5-Minute Check 1
Find x.
A. 1 B. 2 C. 3 D. 4
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5-Minute Check 2
Find x.
A. 1 B. 2 C. 3 D. 4
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5-Minute Check 2
Find x.
A. 1 B. 2 C. 3 D. 4
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5-Minute Check 3
Find x.
A. 2 B. 4 C. 6 D. 8
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5-Minute Check 3
Find x.
A. 2 B. 4 C. 6 D. 8
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5-Minute Check 4
Find x.
A. 10 B. 9 C. 8 D. 7
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5-Minute Check 4
Find x.
A. 10 B. 9 C. 8 D. 7
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5-Minute Check 5
Find x in the figure.
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5-Minute Check 5
Find x in the figure.
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CCSS
Content Standards G.GPE.1 Derive the equation of
a circle of given center and radius using the
Pythagorean Theorem complete the square to find
the center and radius of a circle given by an
equation. G.GPE.6 Find the point on a directed
line segment between two given points that
partitions the segment in a given
ratio. Mathematical Practices 2 Reason abstractly
and quantitatively. 7 Look for and make use of
structure.
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Then/Now
You wrote equations of lines using information
about their graphs.

• Write the equation of a circle.

• Graph a circle on the coordinate plane.

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Vocabulary

• compound locus

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Concept
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Example 1
Write an Equation Using the Center and Radius
A. Write the equation of the circle with a center
at (3, 3) and a radius of 6.
(x h)2 (y k)2 r 2 Equation of
circle (x 3)2 (y (3))2 62 Substitution
(x 3)2 (y 3)2 36 Simplify. Answer
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Example 1
Write an Equation Using the Center and Radius
A. Write the equation of the circle with a center
at (3, 3) and a radius of 6.
(x h)2 (y k)2 r 2 Equation of
circle (x 3)2 (y (3))2 62 Substitution
(x 3)2 (y 3)2 36 Simplify. Answer (x
3)2 (y 3)2 36
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Example 1
Write an Equation Using the Center and Radius
B. Write the equation of the circle graphed to
the right.
The center is at (1, 3) and the radius is 2.
(x h)2 (y k)2 r 2 Equation of
circle (x 1)2 (y 3)2 22 Substitution (x
1)2 (y 3)2 4 Simplify. Answer
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Example 1
Write an Equation Using the Center and Radius
B. Write the equation of the circle graphed to
the right.
The center is at (1, 3) and the radius is 2.
(x h)2 (y k)2 r 2 Equation of
circle (x 1)2 (y 3)2 22 Substitution (x
1)2 (y 3)2 4 Simplify. Answer (x 1)2
(y 3)2 4
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Example 1
A. Write the equation of the circle with a center
at (2, 4) and a radius of 4.
A. (x 2)2 (y 4)2 4 B. (x 2)2 (y 4)2
4 C. (x 2)2 (y 4)2 16 D. (x 2)2 (y
4)2 16
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Example 1
A. Write the equation of the circle with a center
at (2, 4) and a radius of 4.
A. (x 2)2 (y 4)2 4 B. (x 2)2 (y 4)2
4 C. (x 2)2 (y 4)2 16 D. (x 2)2 (y
4)2 16
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Example 1
B. Write the equation of the circle graphed to
the right.
A. x2 (y 3)2 3 B. x2 (y 3)2 3 C. x2
(y 3)2 9 D. x2 (y 3)2 9
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Example 1
B. Write the equation of the circle graphed to
the right.
A. x2 (y 3)2 3 B. x2 (y 3)2 3 C. x2
(y 3)2 9 D. x2 (y 3)2 9
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Example 2
Write an Equation Using the Center and a Point
Write the equation of the circle that has its
center at (3, 2) and passes through (1, 2).
Step 1 Find the distance between the points to
determine the radius.
Distance Formula
(x1, y1) (3, 2) and(x2, y2) (1, 2)
Simplify.
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Example 2
Write an Equation Using the Center and a Point
Step 2 Write the equation using h 3, k 2,
andr 4.
(x h)2 (y k)2 r 2 Equation of
circle (x (3))2 (y (2))2
42 Substitution (x 3)2 (y 2)2
16 Simplify. Answer
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Example 2
Write an Equation Using the Center and a Point
Step 2 Write the equation using h 3, k 2,
andr 4.
(x h)2 (y k)2 r 2 Equation of
circle (x (3))2 (y (2))2
42 Substitution (x 3)2 (y 2)2
16 Simplify. Answer (x 3)2 (y 2)2 16
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Example 2
Write the equation of the circle that has its
center at (1, 0) and passes through (3, 0).
A. (x 1)2 y2 16 B. (x 1)2 y2 16 C. (x
1)2 y2 4 D. (x 1)2 y2 16
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Example 2
Write the equation of the circle that has its
center at (1, 0) and passes through (3, 0).
A. (x 1)2 y2 16 B. (x 1)2 y2 16 C. (x
1)2 y2 4 D. (x 1)2 y2 16
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Example 3
Graph a Circle
The equation of a circle is x2 4x y2 6y
9. State the coordinates of the center and the
measure of the radius. Then graph the equation.
Write the equation in standard form by completing
the square.
x2 4x y2 6y 9 Original equation x2 4x
4 y2 6y 9 9 4 9 Complete the
squares. (x 2)2 (y 3)2 4 Factor and
simplify. (x 2)2 y (3)2 22 Write 3
as (3) and 4 as 22.
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Example 3
Graph a Circle
With the equation now in standard form, you can
identify h, k, and r.
(x 2)2 y (3)2 22
(x h)2 y k2 r2
Answer
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Example 3
Graph a Circle
With the equation now in standard form, you can
identify h, k, and r.
(x 2)2 y (3)2 22
(x h)2 y k2 r2
Answer So, h 2, y 3, and r 2. The center
is at (2, 3), and the radius is 2.
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Example 3
Which of the following is the graph of x2 y2
10y 0?
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Example 3
Which of the following is the graph of x2 y2
10y 0?
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Example 4
Use Three Points to Write an Equation
ELECTRICITY Strategically located substations
are extremely important in the transmission and
distribution of a power companys electric
supply. Suppose three substations are modeled by
the points D(3, 6), E(1, 1), and F(3, 4).
Determine the location of a town equidistant from
all three substations, and write an equation for
the circle.
Understand You are given three points that lie on
a circle.
Plan Graph ?DEF. Construct the perpendicular bise
ctors 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.
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Example 4
Use Three Points to Write an Equation
Solve Graph ?DEF and construct the
perpendicular bisectors of two sides.
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Example 4
Use Three Points to Write an Equation
The center, C, 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.
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Example 4
Use Three Points to Write an Equation
Answer
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Example 4
Use Three Points to Write an Equation
Answer The location of a town equidistant from
all three substations is at (4,1). The equation
for the circle is (x 4)2 (y 1)2 26.
Check 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.
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Example 4
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.
A. (3, 0) B. (0, 0) C. (2, 1) D. (1, 0)
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Example 4
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.
A. (3, 0) B. (0, 0) C. (2, 1) D. (1, 0)
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Example 5
Intersections with Circles
Find the point(s) of intersection between x2 y2
32 and y x 8.
Graph these equations on the same coordinate
plane.
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Example 5
Intersections with Circles
There appears to be only one point of
intersection. You can estimate this point on the
graph to be at about (4, 4). Use substitution to
find the coordinates of this point algebraically.
x2 y2 32 Equation of circle. x2 (x
8)2 32 Substitute x 8 for y. x2 x2 16x
64 32 Evaluate the square. 2x2 16x 32 0
Simplify. x2 8x 16 0 Divide each side by
2. (x 4)2 0 Factor. x 4 Take the square
root of each side.
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Example 5
Intersections with Circles
Use y x 8 to find the corresponding y-value.
(4) 8 4 The point of intersection is (4, 4).
Answer
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Example 5
Intersections with Circles
Use y x 8 to find the corresponding y-value.
(4) 8 4 The point of intersection is (4, 4).
Answer (4, 4)
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Example 5
Find the points of intersection between x2 y2
16 and y x.
A. (2, 2) B. (2, 2) C. (2, 2), (2, 2) D. (2,
2), (2, 2)
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Example 5
Find the points of intersection between x2 y2
16 and y x.
A. (2, 2) B. (2, 2) C. (2, 2), (2, 2) D. (2,
2), (2, 2)
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End of the Lesson