Avon High School
Section: 2.8
ACE COLLEGE ALGEBRA II - NOTES
Distance, Midpoint Formula and Circles
Mr. Record: Room ALC-129
Day 1 of 1
The Distance Formula
Investigation: Suppose you want to find the distance
between the two points on the graph to the right. What
would you do?
The Distance Formula
The distance, d, between the points ( x1 , y1 ) and ( x2 , y2 ) in the
rectangular coordinate system is
The Midpoint Formula
Consider a line segment whose endpoints are
( x1 , y1 ) and ( x2 , y2 ) . The coordinates of the segment’s
midpoint are
Example 1
Note: This formula is merely a different
way to present the Pythagorean Theorem.
Note: This formula is merely a different way
to present the Pythagorean Theorem.
Using the Distance and Midpoint Formulas
For the line segment whose endpoints are (1, 2) and (7, 3) .
a. find the length of the segment
b. find the midpoint of the segment
Circles
Definition of a Circle
A circle is
The Standard Form of the Equation of a Circle
The standard from of the equation of a circle with center ( h, k ) and radius r is
Using the Equation of a Circle
Example 2
a. Find the center and radius of the circle whose equation is ( x  3)2  ( y  1)2  4 .
c. Use the graph to identify the circle’s domain and range.
b. Graph the circle.
y





x

















Investigation: Prior to the Summer 2012 upgrade of the TI-Nspire OS to 3.2, the only way to graph a
circle on a graphing calculator was to solve for y. Solve the equation in Example 2 and discuss its
results. How would you enter the resulting equation into the calculator in order to graph the entire
circle?
Example 3
Converting a Circle’s Equation to Standard From
The relation x 2  y 2  4 x  6 y  23  0 is an equation of a circle.
Use the completing the square technique (twice) to write the equation in standard form.