Obj. 5 Midpoint and Distance

The student will be able to (I can):

• Find the midpoint of two given points.

• Find the coordinates of an endpoint given one endpoint and a midpoint.and a midpoint.

• Find the distance between two points.

The coordinates of a midpoint are the averages of the coordinates of the endpoints of the segment.

C A T

1 3 21

2 2

− += =

-2 2 4 6 8 10

2

4

6

8

10

x

y

• (5, 6)D

O

G

-2

x-coordinate:

y-coordinate:

2 8 105

2 2

+= =

4 8 126

2 2

+= =

midpoint formula

The midpoint M of with endpoints A(x1, y1) and B(x2, y2) is found by

AB

1 12 2M , 2 2

yxx y+ +

A

B

y

y2

●

Maverage of y1 and y2

0

A

x1 x2

y1

average of x1 and x2

Example Find the midpoint of QR for Q(—3, 6) and R(7, —4)

x1 y1 x2 y2

Q(—3, 6) R(7, —4)

21x 3x 7 42

2 2 2

+ += = =−

21 21

yy 6 4+ +=

−= =21 2

1y

2 2

y 6

2

4+ +=

−= =

M(2, 1)

Problems 1. What is the midpoint of the segment joining (8, 3) and (2, 7)?

A. (10, 10)

B. (5, —2)

C. (5, 5)

D. (4, 1.5)

8 2 105

2 2

+= =

3 7 105

2 2

+= =

Problems 2. What is the midpoint of the segment joining (—4, 2) and (6, —8)?

A. (—5, 5)

B. (1, —3)

C. (2, —6)

D. (—1, 3)

4 6 21

2 2

− += =

Problem 3. Point M(7, —1) is the midpoint of , where A is at (14, 4). Find the coordinates of point B.

A. (7, 2)

B. (—14, —4)

C. (0, —6)

D. (10.5, 1.5)

AB

D. (10.5, 1.5)

14 7 7− = 7 7 0− =

( )4 1 5− − = 1 5 6− − = −

Pythagorean Theorem

In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

2 2 2 22 2or b c b(ca a )+ = = +y

x

a

bc

●

●a

22 2c ba= +22c ba= +

22 164 93= + = +25 5= =

●

distance formula

Given two points (x1, y1) and (x2, y2), the distance between them is given by

Example: Use the Distance Formula to find the distance between F(3, 2) and G(-3, -1)

( ) ( )2

1

2

2 2 1d xx y y= − + −

x1 y1 x2 y2

3 2 —3 —13 2 —3 —1

( ) ( )2 2FG 3 3 1 2= − − + − −

( ) ( )2 26 3 36 9= − + − = +

45 6.7= ≈Note: Remember that the square of a negative number is positivepositivepositivepositive.

Problems 1. Find the distance between (9, —1) and (6, 3).

A. 5

B. 25

C. 7

D. 13

( ) ( )( )22d 6 9 3 1= − + − −

( )2 23 4 25 5= − + = =

Problems 2. Point R is at (10, 15) and point S is at (6, 20). What is the distance RS?

A. 1

B.

C. 41

D. 6.5

41

( ) ( )2 2d 6 10 20 15= − + −

( )2 24 5 41= − + =