What is the difference between a new penny and an old quarter? 

Only 4 Gen!uses

Find the midpoint between the points (–2, 3) and (4, 2).

SOLUTION

–2 + 42

( )3 + 2

2,22

( )52,= 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 (Review)

Objective- To solve problems involving the Pythagorean Theorem and Distance Formula.

For Right Triangles Only!

leg

leg

hypotenuse - always oppositethe right angle

For Right Triangles Only!

leg

leg

hypotenusea

b

c

a

b

c2 2 2a b c

Pythagorean Theorem

For Right Triangles Only!

6

8

x

2 2 2a b c 2 2 26 8 x

236 64 x 2100 x

2100 x10 x

x 10 or 10

Solve for x.

6

t15

2 2 2a b c Solve for t.

2 2 2t 6 15 2t 36 225

36 36 2t 1892t 189t 189

t 13.73 21

20 miles

A car drives 20 miles due east and then 45 miles due south. To the nearest hundredth of a mile, how far is the car from its starting point?

45 milesx

2 2 2a b c 2 2 220 45 x

2400 2025 x 22425 x

22425 x

x 2425 x 49.245 97

Finding the Distance Between Two Points using the DISTANCE FORMULA

Using the Pythagorean theorem

(x 2 – x 1) 2 + ( y 2 – y 1) 2 = d 2

THE DISTANCE FORMULA

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) using the distance formula.

d = (x 2 – x 1) 2 + ( y 2 – y 1) 2

= 10

3.16

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