When two segments have the same length, they are said to be congruent segments. If AB = AC Measure...
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Transcript of When two segments have the same length, they are said to be congruent segments. If AB = AC Measure...
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