section 1.9 distance and midpoint formulas; circles...distance and midpoint formulas; circles . the...
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![Page 1: Section 1.9 Distance and Midpoint Formulas; Circles...Distance and Midpoint Formulas; Circles . The Distance Formula 22 2 1 2 1 22 Find the Distance between (-4,2) and (3, -7) x 3](https://reader034.vdocuments.site/reader034/viewer/2022052216/5f27c8176637274ff755fb87/html5/thumbnails/1.jpg)
Section 1.9
Distance and Midpoint Formulas;
Circles
![Page 2: Section 1.9 Distance and Midpoint Formulas; Circles...Distance and Midpoint Formulas; Circles . The Distance Formula 22 2 1 2 1 22 Find the Distance between (-4,2) and (3, -7) x 3](https://reader034.vdocuments.site/reader034/viewer/2022052216/5f27c8176637274ff755fb87/html5/thumbnails/2.jpg)
The Distance Formula
![Page 3: Section 1.9 Distance and Midpoint Formulas; Circles...Distance and Midpoint Formulas; Circles . The Distance Formula 22 2 1 2 1 22 Find the Distance between (-4,2) and (3, -7) x 3](https://reader034.vdocuments.site/reader034/viewer/2022052216/5f27c8176637274ff755fb87/html5/thumbnails/3.jpg)
![Page 4: Section 1.9 Distance and Midpoint Formulas; Circles...Distance and Midpoint Formulas; Circles . The Distance Formula 22 2 1 2 1 22 Find the Distance between (-4,2) and (3, -7) x 3](https://reader034.vdocuments.site/reader034/viewer/2022052216/5f27c8176637274ff755fb87/html5/thumbnails/4.jpg)
2 2
2 1 2 1
2 2
Find the Distance between (-4,2) and (3,-7)
x
3 4 7 2
49 81
130 11.4
x y y
x
y
![Page 5: Section 1.9 Distance and Midpoint Formulas; Circles...Distance and Midpoint Formulas; Circles . The Distance Formula 22 2 1 2 1 22 Find the Distance between (-4,2) and (3, -7) x 3](https://reader034.vdocuments.site/reader034/viewer/2022052216/5f27c8176637274ff755fb87/html5/thumbnails/5.jpg)
Example
Find the distance between (4,-5) and (9,-2).
![Page 6: Section 1.9 Distance and Midpoint Formulas; Circles...Distance and Midpoint Formulas; Circles . The Distance Formula 22 2 1 2 1 22 Find the Distance between (-4,2) and (3, -7) x 3](https://reader034.vdocuments.site/reader034/viewer/2022052216/5f27c8176637274ff755fb87/html5/thumbnails/6.jpg)
The Midpoint Formula
![Page 7: Section 1.9 Distance and Midpoint Formulas; Circles...Distance and Midpoint Formulas; Circles . The Distance Formula 22 2 1 2 1 22 Find the Distance between (-4,2) and (3, -7) x 3](https://reader034.vdocuments.site/reader034/viewer/2022052216/5f27c8176637274ff755fb87/html5/thumbnails/7.jpg)
Find the midpoint of the segment
whose endpoint are (-1,5) and (6,8).
1 6 5 8,
2 2
5 13,
2 2
x
y
Find the midpoint of the segment
whose endpoint are (-1,5) and (6,8).
1 6 5 8,
2 2
5 13,
2 2
(-1,5)
(6,8)
![Page 8: Section 1.9 Distance and Midpoint Formulas; Circles...Distance and Midpoint Formulas; Circles . The Distance Formula 22 2 1 2 1 22 Find the Distance between (-4,2) and (3, -7) x 3](https://reader034.vdocuments.site/reader034/viewer/2022052216/5f27c8176637274ff755fb87/html5/thumbnails/8.jpg)
Example
Find the midpoint of the segment whose
endpoints are (-1,7) and (-5,9).
![Page 9: Section 1.9 Distance and Midpoint Formulas; Circles...Distance and Midpoint Formulas; Circles . The Distance Formula 22 2 1 2 1 22 Find the Distance between (-4,2) and (3, -7) x 3](https://reader034.vdocuments.site/reader034/viewer/2022052216/5f27c8176637274ff755fb87/html5/thumbnails/9.jpg)
Circles
![Page 10: Section 1.9 Distance and Midpoint Formulas; Circles...Distance and Midpoint Formulas; Circles . The Distance Formula 22 2 1 2 1 22 Find the Distance between (-4,2) and (3, -7) x 3](https://reader034.vdocuments.site/reader034/viewer/2022052216/5f27c8176637274ff755fb87/html5/thumbnails/10.jpg)
![Page 11: Section 1.9 Distance and Midpoint Formulas; Circles...Distance and Midpoint Formulas; Circles . The Distance Formula 22 2 1 2 1 22 Find the Distance between (-4,2) and (3, -7) x 3](https://reader034.vdocuments.site/reader034/viewer/2022052216/5f27c8176637274ff755fb87/html5/thumbnails/11.jpg)
Write the standard form of the equation of
the circle with center (-4,1) and radius of 3.
x
y
2 2 2
2 2
( 4) ( 1) 3
( 4) ( 1) 9
x y
x y
Standard
Form
(-4,1)
3
![Page 12: Section 1.9 Distance and Midpoint Formulas; Circles...Distance and Midpoint Formulas; Circles . The Distance Formula 22 2 1 2 1 22 Find the Distance between (-4,2) and (3, -7) x 3](https://reader034.vdocuments.site/reader034/viewer/2022052216/5f27c8176637274ff755fb87/html5/thumbnails/12.jpg)
Find the center and radius of the circle whose
equation is
Graph the equation.
Use the graph to identify the relation’s domain
and range. Why is it a relation and not a
function?
2 2( 3) ( 4) 9x y
x
y
3
(-3,4) Domain: [-6,0]; Range:[1,7]
Center(-3,4); radius=3
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Example
Write the standard form of the equation of the
circle with center at (-2,7) and a radius of 5.
![Page 14: Section 1.9 Distance and Midpoint Formulas; Circles...Distance and Midpoint Formulas; Circles . The Distance Formula 22 2 1 2 1 22 Find the Distance between (-4,2) and (3, -7) x 3](https://reader034.vdocuments.site/reader034/viewer/2022052216/5f27c8176637274ff755fb87/html5/thumbnails/14.jpg)
Example
Find the center and radius of the circle whose equation is
below. Graph the equation. Use the graph to identify the
relation’s domain and range. 2 2( 6) ( 5) 49x y
x
y
![Page 15: Section 1.9 Distance and Midpoint Formulas; Circles...Distance and Midpoint Formulas; Circles . The Distance Formula 22 2 1 2 1 22 Find the Distance between (-4,2) and (3, -7) x 3](https://reader034.vdocuments.site/reader034/viewer/2022052216/5f27c8176637274ff755fb87/html5/thumbnails/15.jpg)
2 2
2 2
If we take the equation from the previous
problem we can multiply out the factors
and move all terms to one side to get the
general form of the equation of the circle.
( 6) ( 5) 49
12 36 10 25
x y
x x y y
2 2
49
12 10 12 0x y x y
General
Form
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2 2
2 2
2 2
2 2
Complete the square and write the
equation in standard form. Then
give the center and radius of each
circle.
x 14 8 29 0
x 14 8 29
(x 14 49)+( 8 16) 29 49 16
x-7 y+4
y x y
x y y
x y y
36
(7,-4); radius=6Center
? ?
![Page 17: Section 1.9 Distance and Midpoint Formulas; Circles...Distance and Midpoint Formulas; Circles . The Distance Formula 22 2 1 2 1 22 Find the Distance between (-4,2) and (3, -7) x 3](https://reader034.vdocuments.site/reader034/viewer/2022052216/5f27c8176637274ff755fb87/html5/thumbnails/17.jpg)
Example
2 2x 4 12 15 0y x y
x
y
Complete the square and write the equation in
standard form. Then give the center and radius of the
circle and graph the equation.
![Page 18: Section 1.9 Distance and Midpoint Formulas; Circles...Distance and Midpoint Formulas; Circles . The Distance Formula 22 2 1 2 1 22 Find the Distance between (-4,2) and (3, -7) x 3](https://reader034.vdocuments.site/reader034/viewer/2022052216/5f27c8176637274ff755fb87/html5/thumbnails/18.jpg)
Example
2 2x 6 8 0y x y
x
y
Complete the square and write the equation in
standard form. Then give the center and radius
of the circle and graph the equation.
![Page 19: Section 1.9 Distance and Midpoint Formulas; Circles...Distance and Midpoint Formulas; Circles . The Distance Formula 22 2 1 2 1 22 Find the Distance between (-4,2) and (3, -7) x 3](https://reader034.vdocuments.site/reader034/viewer/2022052216/5f27c8176637274ff755fb87/html5/thumbnails/19.jpg)
(a)
(b)
(c)
(d)
Find the distance between the points (-1,8)
and (9,5).
55
73
91
109
![Page 20: Section 1.9 Distance and Midpoint Formulas; Circles...Distance and Midpoint Formulas; Circles . The Distance Formula 22 2 1 2 1 22 Find the Distance between (-4,2) and (3, -7) x 3](https://reader034.vdocuments.site/reader034/viewer/2022052216/5f27c8176637274ff755fb87/html5/thumbnails/20.jpg)
(a)
(b)
(c)
(d)
Find the midpoint of the line segment with the
endpoints of (-1,-1) and (- 5,8).
73,
2
92,
2
3 9,
2 2
93,
2
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(a)
(b)
(c)
(d)
Write the equation of the circle with
center at (0,7) and a radius of 4.
2 2
2 2
2 2
2
x 49
(x-7) ( 7) 16
x ( 7) 16
( 7) 16
y
y
y
y