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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