1. The Distance and Midpoint Formulas
By Tristen Billerbeck

2. Ruler Postulate Rules that are accepted without proof are postulates.
Points on a line can be matched to real numbers or coordinates.
The distance or length between two points A and B is written AB.
Distance is the absolute value of the difference between the coordinates. A B
Name
Coordinate

3. Example 1:
Find the length of segment AB A B

4. Example 2:
Find the length of segment AB A B

5. The Distance Formula
If A and B are points in a coordinate plane, then the distance between A and B is: B c b A a C

6. Example 1:
Given A(3,7) and B(-1,4), find the length of AB:

7. Example 2:
Given A(-2,7) and B(3,6), find the length of AB:

8. Vocabulary
Midpoint: the point that divides a segment into two congruent (=) segments.
Bisect: to divide into two equal parts (bi-)
Segment Bisector: a segment, ray, line, or plane the intersects a segment at its midpoint. ~ M B A
Congruency Mark
Midpoint

9. ( , ) The Midpoint Formula Average y (x2, y2) (x1, y1) Average x
If A(x , y ) and B(x , y ) are points in a coordinate plane, then the midpoint of AB has coordinates:
( , ) 1 1 2 2
Average y
(x2, y2)
(x + x )
(y + y ) B 2 1 2 1 2 2 M
(x1, y1)
|y – y | 2 1 A C
|x – x | 2 1
Average x

10. Example 1
Find the midpoint of AB if A(6, 4) and B(-2, 4).
( , ) = (4/2, 8/2)
= (2, 4)
( )
( ) 2 2

11. Example 2
Find the midpoint of AB if A(-3, -1) and B(7, 1).
( , ) = (4/2, 0/2)
= (2, 0)
( )
( ) 2 2