Distance and Midpoint

Sec: 1.3G.2b&c, G.11b

Midpoint

Is a point in a line segment that splits the line into two congruent segments.

Therefore, AX=XB

A

Midpoint

BX

Segment Bisector

Is a segment bisector of

A BM

is a point, ray, line, line segment or plane that intersects a segment as a midpoint.

D

C

CDAB

MBAM Therefore, and MBAM

Ex:

In the skateboard design, BISECTS at point T, and find .

VW XY

cmXT 9.39 XY

X 39.9

T

YW

V

TYXT TYXTXY

9.399.39 XYcmXY 8.79

Using Algebra with line segments

Point M is the midpoint of ; Find the length of .

V WM

VW

VM

4x - 1 3x + 3

Try

P QM

l

PQPQ

Identify the segment bisector of , then find the measure of

5x - 7 11-2x

The Midpoint Formula

• The coordinates of a segment are the averages of the x-coordinates and of the y-coordinates of the endpoints.

• If A(x1, y1) and B(x2, y2) are points in a coordinate plane, then the midpoint M of AB has coordinates:

• Diagram on overhead.

2,

22121 yyxx

Example:

Find the Midpoint if the endpoints of are

R(1, -3) and S(4, 2).

RS

Try This

Find the Midpoint(M) of if the endpoints are

A(1,2) and B(7, 8)

AB

Ex: Finding the missing endpoint

Find the coordinates of the missing endpoint of when M(2,1) and one endpoint is J(1,4). Find

the coordinates of K. JK

Try this:

Find the coordinates of the missing endpoint of when M(-1,-2) and one endpoint is W(4,4).

Find the coordinates of V. VW

Distance formula

Computes the distance between two points on a coordinate plane.If A(x1, y1) and B(x2, y2) are points in the coordinate plane, then the distance between A and B is:

Diagram on Overhead.

2122

12 yyxxAB

Example

What is the Approximate length of with endpoints R(2, 3) and S(4, -1).

RS

Try this

Find the length of with endpoints A(-3, 2) and B(1, -4)

AB