lesson 10-4
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LESSON 10-4. Equations of Circles. Created by Lisa Palen and Kristina Green Henrico High School. Part I. Equations of Circles. Recall: Definitions. Circle: The set of all points that are the same distance from the center - PowerPoint PPT PresentationTRANSCRIPT
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Equations of Circles
LESSON 10-4
Created by Lisa Palen and Kristina GreenHenrico High School
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Equations of Circles
Part I
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Recall: Definitions
• Circle: The set of all points that are the same distance from the center
• Radius: a segment whose endpoints are the center and a point on the circle
• Radius: the LENGTH of a radius
RadiusCenter
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Equation of a Circle2 2 2x y r Center (0, 0)
Radius = r
Center (h, k)Radius = r 2 2 2x h y k r
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Finding the Center and the Radius when given the equation
2 2
2 2
2 2
2 2
22
2 2
1. 25
2. 100
3. 5 4 49
4. 7 3 3
5. 1 12
6. 3 81
x y
x y
x y
x y
x y
x y
Center (0, 0), r = 5Center (0, 0), r = 10
Center (5, -4), r = 7
Center (-7, 3), r =
Center (0, 1), r =
Center (3, 0), r = 9
3
12
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Writing the Equation of a Circle
1. Center (0, 0) r = 2
2. Center (0, 1) r = 6
3. Center (-3, 5) r = 2.5
4. Center (-5, 10) r = 10
5. Center (8, 0) r = 1
6. Center (6, 9) r = 3.4
x2 + y2 = 4x2 + (y – 1)2 = 36(x + 3)2 + (y– 5)2= 6.25(x + 5)2 + (y–10)2= 100(x – 8)2 + y2= 1(x– 6)2 + (y– 9)2= 11.56
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Writing the Equation of a circle2. A circle whose center is at (-3, 2) passes through (-7, 2).
a. What is the length of the radius of the circle?
b. Write the equation of the circle.
Answers: a. r = 4 b. (x + 3)2 + (y - 2)2 = 16
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Graphing a CircleFind the center and the radius and graph the circle.
2 2 9x y
Answers: center (0, 0) radius = 3
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Graphing a CircleFind the center and the radius and graph the circle.
2 21 2 25x y
Answers: center (1, -2) radius = 5
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Graphing a CircleFind the center and the radius and graph the circle.
2 23 4x y
Answers: center (3, 0) radius = 2
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Writing the Equation of a circle3. A circle has a diameter with endpoints
A (1, 2) and B (3, 6).
a. What is the center of the circle?
b. What is the radius of the circle?
c. What is the equation of the circle?Answers: a. (2, 4) b. sqrt (5) c. (x – 2)2 + (y – 4)2 = 5
The midpoint of segment AB!
The distance from the center to A or B!
diam
eter
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Finding the midpoint
For the last problem it was necessary to find the midpoint, or the point halfway between two points. There is a formula for this.
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Midpoint
Part II
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Reminder: What is a Midpoint?• The midpoint of a segment AB is the point that divides
AB into two congruent segments.• Where is the midpoint of AB?
A
BOver Here
?
Over Here
?
Over Here
?
Here it is!
midpoint
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Midpoint on a Number Line
• To find the midpoint of two points on a number line, just average the coordinates.
• Find the midpoint of GT.
a b
2
x
• Take the average of the coordinates:
G T
4 9
2
5
2 = 2.5
midpoint
a b
2
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Finding a Midpoint inThe Coordinate Plane
x
y
We can find the midpoint between any two points in the coordinate plane by finding the midpoint of the x-coordinates and the midpoint of the y-coordinates.
midpoint?
Example Find the midpoint of the two points.
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Finding a Midpoint inThe Coordinate Plane
a b 2 3 10.5
2 2 2
x
y
First: Find the average (midpoint) of the x-coordinates.Remember: Take the average of the two coordinates.
– 4
8
average of x-coordinates
a b 4 8 42
2 2 2
2
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Finding a Midpoint inThe Coordinate Plane
a b 2 3 10.5
2 2 2
x
y
Next: Find the midpoint (average) of the y-coordinates.Remember: Take the average of the two coordinates.
– 2
3
a b 2 3 10.5
2 2 2
average of y-coordinates
average of x-coordinates
0.5
2
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Finding a Midpoint inThe Coordinate Plane
a b 2 3 10.5
2 2 2
x
y
midpoint of y-coordinates
midpoint of x-coordinates
0.5
2
Finally: The midpoint is the ordered pair:
(average of x-coordinates, average of y-coordinates)
= (2, 0.5)
(2, 0.5)
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The Midpoint FormulaThe following formula combines what we did:
midpoint = (average of x-coordinates, average of y-coordinates)
where (x1, y1) and (x2, y2) are the ordered pairs corresponding to the two points.
So let’s go back to the example.
1 2 1 2x x y y,
2 2
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Example
x
y
Find the midpoint of the two points.
Solution: We already know the coordinates of the two points.
(– 4, – 2)
(8, 3)midpoint?
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Example cont.
1 2 1 2x x y y,
2 2
4 8 2 3,
2 2
4 1,
2 2
Solution cont.
Since the ordered pairs are
(x1, y1) = (-4, -2) and (x2, y2) = (8, 3)
Plug in x1 = -4, y1 = -2, x2 = 8 and y2 = 3 into
midpoint =
=
=
= (2, 0.5)
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THINK ABOUT IT
Find the center, the length of the radius, and write the equation of the circle if the endpoints of a diameter are (-8,2) and (2,0).
Center: Use midpoint formula!
Length: use distance formula with center and an endpoint
8 2 2 0,
2 2
3,1 2 2(2 ( 3)) (0 1) 26
Equation: Put it all together
22 2( 3) ( 1) 26x y or 2 23 ( 1) 26x y