12.7
The Distance Formula
How far was the soccer ball kicked?
Goal
Find the distance
between two points in a
coordinate plane.
You can use the distance formula to
find the distance between two points
in a coordinate plane. In Example 3
you will find the distance that a
soccer ball was kicked.
Key Words
• distance formula
To find a general formula for the distance
between two points A(x1, y1) and B(x2, y2),
draw a right triangle as shown at the right.
Using the Pythagorean theorem, you can
write the equation
y
d
A (x1, y 1)
x2 x1
(x2 x1) (y2 y1) d .
2
2
B (x2, y2)
2
y2 y1
C (x2, y1)
x
Solving the equation for d leads to the
following distance formula.
THE DISTANCE FORMULA
The distance d between the points (x 1, y1) and (x2, y2) is
d
EXAMPLE
1
2
2
x
y
.
(x
(y
2
1)
2
1)
Find the Distance Between Two Points
Use the distance formula to find the distance between (1, 4) and (2, 3).
x
) 2
( y2
y
)2
d (x
2
1
1
730
Chapter 12
Write distance formula.
(
2
)
12
(3
)
42
Substitute.
(
3)2
(
1)2
Simplify.
9
1
Evaluate powers.
10
Add.
≈ 3.16
Use a calculator.
Radicals and More Connections to Geometry