Distance and Midpoints

Objective:

(1)To find the distance between two points(2) To find the midpoint of a segment

Definitions

• Midpoint: The points halfway between the endpoints of a segment.

• Distance Formula: A formula used to find the distance between two points on a coordinate plane.

• Segment Bisector: A segment, line, or plane that intersects a segment at its midpoint.

Midpoint

• To find the midpoint along the number line, add both numbers and divide by 2.

2

84

40 82 6 10 12-2-4-6

A B C D E F G H I J

Find the midpoint of BH

2

ba 22

4

The coordinate of the midpoint is 2.

E is the midpoint.

More Midpoint

• For the midpoint on a coordinate plane, the formula is:

2,

22121 yyxx

M

B(-1,7)

A(-8,1)

2

71,

2

)1(8

2

8,

2

9

4,

2

14This is the midpoint.

Finding the endpoint of a segment

• We’re still going to use the Midpoint Formula:

• But the there will be a few unknowns:

2,

22121 yyxx

M

Find the coordinates for X if M(5,-1) is the midpoint and the other endpoint has coordinates Y(8,-3)

• helps us find the x-coordinate of the endpoint.

221 xx

Finding the endpoint of a segment

52

8 2 x

522

82 2

x

108 2 x

Multiply both sides by 2 to eliminate the denominator

-8 -8 Subtract 8 from both sides

x2 = 2 This is the x-coordinate of the other endpoint

221 yy This helps us find

the y-coordinate of the midpoint

12

3 2 y

122

32 2

y

Finding the endpoint of a segment

23 2 y+3 +3

y2 = 1 This is the y-coordinate of the endpoint

The coordinate of the other endpoint is X(2,1).

Finding the value of a variable

M is the midpoint of AB.

Find the value of x:

Since M is a midpoint, that means that AM=MB which means

3x – 5 = x + 9

-x -x

2x – 5 = 9

+5 +5

2x = 14

A

M

B

3x - 5

x + 9

2x = 14

2 2

x = 7

Distance

• Remember:

AB means the length of AB

To find the distance on the number line, take the absolute value of the difference of the coordinates.

a – b

0 82 6 10 12-2-4-6 4

A B C D E F G H I J

Find CJ

-2 -12=-14= 14

CJ = 14

Find EA2 – (-6)=2+6=8= 8EA = 8

More Distance

The distance between two points in the coordinate plane is found by using the following formula:

212

212 )()( yyxxd

A(-3,1)

B(4,-2)

22 )12()]3(4[ d

22 )3()7( d

949d58d6.7d