9.1 Apply the Distance and Midpoint Formulas

Distance formula

The formula used to find the distance between two points A(x_{1 ,} y₁) and B(x₁, y₂)

Midpoint formula

The formula that describes the midpoint of the line segment joining A(x_{1 ,} y₁) and B(x₂, y₂)

THE DISTANCE FORMULA

The distance d between (x₁, y₁)and(x₂, y₂) is

Example 1

Find the distance between two points

Find the distance between (-5, -3) and (3, 6).

Let (x_{1 ,} y₁) = (-5, -3) and (x₂, y₂) = (3, 6).

Complete the following exercises.

1. Find the distance between (-7, 3) and (5, -2).

13

2. The vertices of a triangle are T(2, 1), U(4, 6), and V(7, 3). Classify DTUV as scalene, isosceles, or equilateral.

isosceles

THE MIDPOINT FORMULA

A line segment's midpoint is __equidistant__ from the segment's endpoints. The midpoint formula describes the __midpoint__ of a line segment joining A(x_{1 ,} y₁) and B(x₂, y₂) as follows:

In words, each coordinate of M is the __mean__ of the corresponding coordinates of A and B.

Example 3

Find the midpoint of the line segment joining (-6, 5) and (2, -3).

Let (x₁, y₁) = (-6, 5) and (x₂, y₂) = ( 2, -3).

= (_-2__,_1_)

Example 4

Find a perpendicular bisector

Write an equation for the perpendicular bisector of the line segment joining A(-4, 1) and B(2, 3).

Solution

1. Find the midpoint of the line segment.

2. Calculate the slope of.

3. Find the slope of the perpendicular bisector.

4. Use point-slope form: y - _2_ = -3_ (x - (_-1_)) or y = __-3x - 1__ .

An equation for the perpendicular bisector of AB is y = __-3x - 1__.