1. Congruent segments MIDPOINT OF A SEGMENT
1-3 Distance & Midpoints
Congruent segments
MIDPOINT OF A SEGMENT

2. A C
10 cm
10 cm B D

3. Definition of Congruent segments
“Two segments are said to be congruent if and only if they have the same measure.” There is a phrase “if & only if or Iff” which means that the definition is two way or (bi-conditional). 1) If the segments are congruent, then they are equal. 2) If the segments are equal, then they are congruent.

4. Distance on a Number Line
The distance between points A and B written as AB, is the absolute value of the difference between the coordinates of A and B (a , b).
AB is also called the measure of AB. A B a b
AB = | a – b | or | b – a |

5. Calculating Distance in the Coordinate PlaneA B

6. Example 4:

7. Bisector of a Segment F l G M H

8. Definition of a Midpoint of a SegmentRS MS

9. Postulate: Midpoint Postulate
A segment has exactly one midpoint
If M is the midpoint of AC, AM= 3x +2 and MC = 2x + 6, find x.
SOLUTION:
Write an equation for x using definition of congruent segments, then solve.
AM = MC ( M is the midpoint of AC)
3x + 2 = 2x + 6
3x – 2x= 6- 2
x = 4

10. Example 1: 2x + 8 = 5x + 2 8 – 2 = 5x – 2x 6 = 3x 2 = x
Points D, E & F are the three collinear points and point E is a midpoint. If DE = FE and DE = 2x + 8, FE = 5x + 2, find x.
SOLUTION:
Write an equation for x using definition of congruent segments, then solve.
DE = FE ( GIVEN)
2x + 8 = 5x + 2
8 – 2 = 5x – 2x
6 = 3x
2 = x
D E F
2x x +2

11. AM = MC ( M is the midpoint of AC)
Example 2:
If M is the midpoint of AC, AM= 3x +2 and MC = 2x + 6, find x.
SOLUTION:
Write an equation for x using definition of congruent segments, then solve.
AM = MC ( M is the midpoint of AC)
3x + 2 = 2x + 6
3x – 2x= 6- 2
x = 4

12. Formula: The Coordinate of the Midpoint of a Segment on a Number Line

13. Coordinate of point E = (D + F) ÷ 2
Example 3:
Find the coordinate of point E, if point E is the midpoint of DF
SOLUTION:
Coordinate of point E = (D + F) ÷ 2
Coord. Point E = (-5 + 9) ÷ 2
Coord. Point E= 4 ÷ 2
Coord. Point E= 2
D E F 2

14. Formula: The Coordinates of the Midpoint of a Segment in the Coordinate Plane

15. Example 5:
Given the endpoints (x,,y1) and (x2, y2) of the Line segment we can use the formula
Let the endpoints be (2, 8) and (6, 2) =

16. Midpoint of AB with midpoint M(5, 6) and endpoint B (6,4)
Example 6:
Midpoint of AB with midpoint M(5, 6) and endpoint B (6,4)
6 + x = y = 12
x = and y = 8
Endpoint A(4,8)

17. ASSIGNMENT
Pg #’s 1-31 all even 64,65,81