51 Midsegment Theorem and Coordinate Proof

1
5-1 Midsegment Theorem and Coordinate Proof

• Goals
• To use properties of midsegments and write
  coordinate proofs.

2
Terms

• A midsegment of a triangle is the segment
  connecting the midpoints of two sides.
• A coordinate proof involves placing geometric
  figures in a coordinate plane.

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Theorem 5-1 Midsegment Theorem

• If a segment joins the midpoints of two sides of
  a triangle, then the segment is parallel to the
  third side and is half its length.
• DE // AC
• DE ½ AC

B
E
D
A
C
4
Example
In ?XYZ, M, N, and P are midpoints. The
perimeter of ?MNP is 60. Find NP and YZ, and the
perimeter of ?XYZ. YZ 2(22) NP 60
22 24 XY 2(14) XZ
2(24) P?XYZ 2(60)
44
48
28
14
28
14
48
44
120
5
Example

• Sketch ?ABC. Find the length and the slope of
  each side if A(0, n), B (m, n), and C(m, 0).
  Then find the coordinates of each midpoint. Is
  ?ABC a right triangle? Is it isosceles?
  Explain. (All variables are positive, m ?n)

BC n slope und. mdpt
AB m slope 0 mdpt
Right ?, Not Isosceles
A(0, n)
B(m, n)
AC slope mdpt
C(m, 0)