Distance,

Midpoints &

Circles Mr. Velazquez

October 8,

2015

Honors Precalculus 1

The

Distance

Formula

October 8, 2015

Honors Precalculus

2

The Distance Formula

October 8, 2015 3

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

The Distance Formula

October 8, 2015 4

Find the distance between 4, −5 and 9,−2 :

The Midpoint Formula

October 8, 2015 5

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) Find the midpoint of the segment

whose endpoints are −1,5 and 6,8 :

−1 + 6

2,5 + 8

2

5

2,13

2

The Midpoint Formula

October 8, 2015 6

Find the midpoint of the segment whose endpoints are −1,7

and −5, 9 :

Classwork/Exit Ticket

October 8, 2015 7

A line segment, 𝐿 , extends from the point

−3,−1 to the point 7, 5 . Use the

distance and midpoint formulas to find

the length of 𝐿 and the coordinates of

the midpoint of 𝐿 .

Circles

October 8, 2015 8

Graphing Circles

October 8, 2015 9

Since the equation representation of a circle is not a function, to

graph a circle using a graphing utility, we must consider two

separate functions when solving for y:

𝑥2 + 𝑦2 = 4

𝑦2 = 4 − 𝑥2

𝑦 = ± 4 − 𝑥2

𝑦 = + 4 − 𝑥2 and 𝑦 = − 4 − 𝑥2

So we graph both the positive and negative versions of y

separately, which together will form a circle.

Circles

October 8, 2015 10

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

Circles

October 8, 2015 11

Find the center and radius of the circle whose

equation is 𝑥 + 3 2 + 𝑦 − 4 2 = 9

Graph the equation.

Use the graph to identify the

relation’s domain and range. Why is

it a relation and not a function?

x

y

3

(-3,4)

Center: (−3, 4) Radius: 3

Domain: [−6, 0]; Range: [1, 7]

Circles

October 8, 2015 12

Write the standard form of the equation of the circle with center

(−2, 7) and radius of 5.

Circles

October 8, 2015 13

Find the center and radius of the circle represented by the equation

𝑥 − 6 2 + 𝑦 + 5 2 = 49. Find the domain and range and graph it.

x

y

Circles, General Form

October 8, 2015 14

Using the previous equation, we can multiply out the factors

and move all terms to the left side, combining like terms to

obtain the general form of the equation.

𝑥 − 6 2 + 𝑦 + 5 2 = 49

𝑥2 − 12𝑥 + 36 + 𝑦2 + 10𝑦 + 25 = 49

𝑥2 + 𝑦2 − 12𝑥 + 10𝑦 + 12 = 0

General Form

Circles, General Form

October 8, 2015 15

Complete the square and write the equation in standard form.

Then give the center and radius of the circle.

𝑥2 + 𝑦2 − 14𝑥 + 8𝑦 + 29 = 0

𝑥2 − 14𝑥+ ? +𝑦2 + 8𝑦+ ?= −29

𝑥2 − 14𝑥 + 𝟒𝟗 + 𝑦2 + 8𝑦 + 𝟏𝟔 = −29 + 𝟒𝟗 + 𝟏𝟔

𝑥 − 7 2 + 𝑦 + 4 2 = 36

Center: (7, −4); Radius= 6

Example

October 8, 2015 16

Complete the square and write the equation in standard form. Then

give the center and radius of the circle and graph the equation.

x

y 2 2x 4 12 15 0y x y

Example

October 8, 2015 17

Complete the square and write the equation in standard form. Then

give the center and radius of the circle and graph the equation.

x

y2 2x 6 8 0y x y

Classwork/Exit Ticket

October 8, 2015 18

You are trying to draw the graph of a dartboard on a

coordinate plane. The design of this dartboard is simple: a total

of four concentric (same center) circles, where the radius of the

outer circle is 1 unit longer than the radius of the next largest

circle, whose radius is 1 unit longer than the next, etc. The area

of the entire dartboard (i.e. the outer circle) should be at least

25𝜋, and the center of the dartboard should be located

somewhere in the first quadrant.

Choose a center and outer radius that fits the requirements,

and write four equations (in standard form) to represent each of

the four circles needed to draw your dartboard. Then graph

your dartboard on a sheet of graph paper.