When two segments have the same length, they are said to be congruent segments.

If AB = AC

Measure of segments

Congruent

Segments

then AB = ACA

B C

Is read: “ Segment AB is congruent to segment AC”

Is read: “ The measure of segment AB is equal to the measure of segment AC”

Do modeling activity on page 36: Locating the midpoint of a segment

You can also compare the measure of segments. For example you can say: AB < BC or BC > AC

Congruent Segments

The midpoint of a segment is the point equidistant from the endpoints of the segment.

Definition of Midpoint: The midpoint M of PQ is the point between P and Q such that PM=MQ

P QM

MIDPOINTS AND SEGMENT CONGRUENCE

. . .

On the number line, the coordinates of the midpoint of a segment whose endpoints have coordinates a and b is a+b 2 In a coordinate plane, the coordinates of the midpoint of a segment whose endpoints have coordinates (x1, y1) and (x2, y2) are (x1+x2, y1+y2 )

2 2

Midpoint Formulas

Example 1 (p.37)

a) Use the number line to find the coordinates of the midpoint of FG

b) Find the coordinates of Q, the midpoint of RS, if the endpoints of RS are R (-3,-4) and S (5,7).

F G

-3 –2 –1 0 1 2 3 4 5 6 7 8

0

Examples

Example 2: p. 38 (Students do then I explain) Find the coordinates of point Q if L(4,-6) is the midpoint of NQ and the coordinates of N are (8,-9)

If Y is the midpoint of XZ, XY = 2a + 11, and YZ = 4a –5, find the value of a and the the measure of XZ

More examples

Segment bisector: is a segment line or plane that intersects a segment at its midpoint.

P QM

T.

... .R

N

M

M, TM, RM and plane N are all bisectors of PQ

Do construction p. 39

Segment Bisector

. .

Proof of theorem 1-1: Given that M is the midpoint of AB, write a paragraph proof to show that AM = MB.

(Theorem 1-1) Midpoint theorem: If M is the midpoint of AB then AM = MB

The Midpoint Theorem

A proof: is a logical argument in which each statement you make is backed up by a statement that is accepted as true.

Paragraph proof or informal proof: a paragraph that explains why a conjecture for a given situation is true.

Conjecture: is an educated guess.

What are proofs?

From the definition of midpoint of a segment, we know that AM = MB.

A.

B .

M.

That means that AM and MB have the same measures

By the definition of congruence, if AM and MB have the same measure, they are congruent segments.

Thus, AM = MB.

Paragraph proof of theorem 1-1