yB – yA

xB – xA

THE DISTANCE AND MIDPOINT FORMULAS

Investigating Distance:

2 Find and label the coordinates of the vertex C.

3

x

y

C (6, 1)

1 Plot A(2,1) and B(6,4) on a coordinate plane. Then draw a right triangle that has AB as its hypotenuse.

Remember:

a 2 + b 2 = c 2

4 – 1

6 – 2

AB

4

3

4 2 + 3 2 = c 2

16 + 9 = c 2

16 + 9 = c

25 = c

5

4 Use the Pythagorean theorem to find AB.

AB = 5

B (6, 4)

A (2, 1)

Find the lengths of the legs of ABC.

Finding the Distance Between Two Points

Using the Pythagorean theorem

(x 2 – x 1) 2 + ( y 2 – y 1) 2 = d 2

THE DISTANCE FORMULA

The distance d between the points (x 1, y 1) and (x 2, y 2) is

d = (x 2 – x 1) 2 + ( y 2 – y 1) 2

Solving this for d produces the

distance formula.

You can write the equation

a 2 + b 2 = c 2

x 2 – x 1

y2 – y1

d

x

y

C (x 2, y 1 )

B (x 2, y 2 )

A (x 1, y 1 )

The steps used in the investigation can be used to develop a general formula for the distance between two points A(x 1, y 1) and B(x 2, y 2).

Finding the Distance Between Two Points

Find the distance between (1, 4) and (–2, 3).

d = (x 2 – x 1) 2 + ( y 2 – y 1) 2

= 10

3.16

To find the distance, use the distance formula.

Write the distance formula.

Substitute.

Simplify.

Use a calculator.

SOLUTION

= (x 2 – x 1) 2 + ( y 2 – y 1) 2–2 – 1 3 – 4

Applying the Distance Formula

A player kicks a soccer ball that is 10 yards from a sideline and 5 yards from a goal

line. The ball lands 45 yards from the same goal line and 40 yards from the same

sideline. How far was the ball kicked?

The ball is kicked from the point (10, 5),

and lands at the point (40, 45). Use the

distance formula.

d = (40 – 10) 2 + (45 – 5) 2

= 900 + 1600 = 2500 = 50

The ball was kicked 50 yards.

SOLUTION

Finding the Midpoint Between Two Points

The midpoint of a line segment is the point on the segment that is equidistant from

its end-points. The midpoint between two points is the midpoint of the line segment

connecting them.

THE MIDPOINT FORMULA

The midpoint between the points (x 1, y 1) and (x 2, y 2) isx 1 + x 2

2( )y 1 + y 2

2,

Find the midpoint between the points (–2, 3) and (4, 2). Use a graph to check the

result.

SOLUTION

–2 + 42( )3 + 2

2,22( )5

2,= 1( )52,=

The midpoint is , . 1( )52

x 1 + x 2

2( )y 1 + y 2

2,Remember, the midpoint formula is .

Finding the Midpoint Between Two Points

Find the midpoint between the points (–2, 3) and (4, 2). Use a graph to check the

result.

CHECK

From the graph, you can see that the

point , appears halfway

between (–2, 3) and (4, 2). You can also

use the distance formula to check that

the distances from the midpoint to each

given point are equal.

( )152

Finding the Midpoint Between Two Points

(1, )52

(–2, 3)

(4, 2)

Applying the Midpoint Formula

You are using computer software to design a video game. You want to place a

buried treasure chest halfway between the center of the base of a palm tree and

the corner of a large boulder. Find where you should place the treasure chest.

SOLUTION

Assign coordinates to the locations of the two landmarks. The center of the palm tree is at (200, 75). The corner of the boulder is at (25, 175).

Use the midpoint formula to find the point that is halfway between the two landmarks.

1

2

25 + 2002( )175 + 75

2,225

2( )2502,= = (112.5, 125)

(25, 175)

(200, 75)

(112.5, 125)

Two news helicopters are flying at the same altitude on their way to a political rally. Helicopter A is 20 miles due west of the rally. Helicopter B is 15 miles south and 15

miles east of the rally. How far apart are they?

How far from the rally is each helicopter?Both helicopters are flying at an average of 80mi/hr.

How many minutes will it take each of them to arrive at the scene?

Answer:  a. They are about 30 miles apart b. Helicopter A is 20 miles away. Helicopter B is about 21.2 miles away. c. It will take Helicopter A 15 minutes and Helicopter B about 16 mins.

Compare and Contrast the Distance Formula and the Pythagorean Theorem.

Describe the common mistakes that students may make if they are not careful when using both the midpoint formula and the distance formula.