distance, midpoint, and circles

Distance, Midpoint,

and CirclesLESSON 12.1

Objective

Use distance formula

Use midpoint formula

Write standard form of a circle

Graph a circle

Find center and radius of a circle from an

equation

Distance Formula

The distance of any two points (𝑥1, 𝑦1) and 𝑥2, 𝑦2can be calculated using the distance formula.

𝑥2 − 𝑥12 + 𝑦2 − 𝑦1

2

NOTE: distance is never negative

Distance Formula

Find the distance between the two points.

A) (−3, 6) and (3, −2) B) (−4,−1) and (2, 2)

Distance Formula

Find the following distances using the following

points. 𝐴(−2, 5) 𝐵(12, 3) 𝐶(10,−11)

C) 𝐴𝐵 D) 𝐵𝐶 E) 𝐶𝐴

Midpoint Formula

The midpoint of any two points (𝑥1, 𝑦1) and 𝑥2, 𝑦2can be calculated using the midpoint formula.

(𝑥1 + 𝑥2

2,𝑦1 + 𝑦2

2)

Midpoint Formula

Find the midpoint of the two points.

F) (−3, 6) and (3, −2) G) (−4,−1) and (2, 2)

Circles

A circle is the set of all points that are a fixed distance 𝑟 (radius) from a center (ℎ, 𝑘).

STANDARD FORM

𝑥 − ℎ 2 + 𝑦 − 𝑘 2 = 𝑟2

NOTE: Since both ℎ and 𝑘 are inside parenthesis, BOTH signs must be changed when using the center.

Circles

Write the standard form of the equation of the circle

H) 𝑟 = 3, 𝐶(−2, 5) I) 𝑟 = 5, 𝐶(−1,−3)

J) 𝑟 =1

2, 𝐶(0,3)

Circles

K) Write the standard form of the equation of the

circle with center at the origin and containing the

point (4,6)

Circles

To graph a circle

1. Identify the center (ℎ, 𝑘) and the radius 𝑟

2. Plot the center. From the center go up, down,

left, and right the distance 𝑟

3. Connect the 4 outside points as a circle

Circles

Graph the equation. Give the domain and range.

L) 𝑥 + 3 2 + 𝑦 − 2 2 = 36

Circles

Graph the equation. Give the domain and range.

M) 𝑥 − 6 2 + 𝑦2 = 16

Circles – Change Form

It is possible for an equation to not be in the needed

form. Using completing the square, we can change the form from general form to standard form.

General form: 𝑥2 + 𝑦2 − 24𝑥 − 12𝑦 + 172 = 0↓

Standard form: 𝑥 − 12 2 + 𝑦 − 6 2 = 8

Circles – Change Form

To change the form of the circle to standard form

1. Rearrange the terms

matching variables together with constant on other side

2. Compete the square – keep things balanced

twice: once for 𝑥 and once for 𝑦

3. Each set of C.T.S. should be factored

Circles – Change Form

N) Change the circle from general to standard form.

𝑥2 + 𝑦2 + 4𝑥 − 32𝑦 + 256 = 0

Circles – Change Form

O) Change the circle from general to standard form.

𝑥2 + 𝑦2 + 16𝑥 + 2𝑦 − 35 = 0