Transcript Factors, Fractions, and Exponents

Midsegments of
Triangles
5.1
Today’s goals
By the end of class today, YOU should be able to…
1. Define and use the properties of midsegments to solve
problems for unknowns.
2. Use the properties of midsegments to make statements
about parallel segments in a given triangle.
3. Understand and write coordinate proofs.
Midsegments

A midsegment of a triangle is a segment
connecting the midpoints of two sides of a
triangle.
Triangle 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.
Proving the Triangle
Midsegment Theorem…

To prove the Triangle Midsegement Theorem,
use coordinate geometry and algebra. This style
of proof is called a coordinate proof. To begin
the proof:
 Place the triangle in a convenient spot on the
coordinate plane.
 Choose variables for the coordinates of the
vertices.
Coordinate proofs cont…
State what you are given.
 State what you want to prove.
 Use the midpoint formula to find the midpoints of
two sides of the triangle.
 To prove that the midsegment and third side are
parallel, show that their slopes are equal.
 To prove that the midsegment is half the length
of the third side, use the distance formula to
calculate the length of both segments.

*See section 5.1 for a complete version of a coordinate proof for the
Triangle Midsegment Theorem
Ex.1: Find the lengths
Using the following illustration (where H, J,
and K are the midpoints), find the lengths
of HJ, JK, and FG:
Ex.1: Solution
HJ = ½ EG = ½ (100) = 50
JK = ½ EF = ½ (60) = 30
HK = ½ FG
FG = 2 HK = 2 (40) = 80
Use the Triangle Midsegment
Theorem to show that the
midsegment = ½ the length of
the third side
You Try…
Use the following triangle to solve the
questions below (A, B, & C are midpoints):
1. DF=120, EF=90, BC=40. Find AB, AC, & DE.
2. EF=10x, AB=4x, AC=10, DE=12x. Find BC & AC.
Ex.2: Identify the parallel
segments
If A, B, & C are midpoints, which segment
is parallel to AC?
Ex.2: Solution
By the Triangle Midsegment Theorem,
AC ll EF
You Try…
If A, B, & C are midpoints:
1. Which segment is parallel to DE?
1. Which segment is parallel to AB?
Practice
Solve with a partner
of your choice…
The triangular face
of the Rock and Roll
Hall of Fame in Cleveland, Ohio is isosceles.
The length of the base is 229 ft 6 in. What is
the length of a segment located half way up
the face of the Rock and Roll Hall of Fame
Explain your reasoning.
Homework
 Page
246 #s 2, 4, 6, 7, 13, 30
 Page 247 #s 26, 34, 35
 The
assignment can also be found
online at:
• http://www.pearsonsuccessnet.com/snpap
p/iText/products/0-13-037878-X/Ch05/0501/PH_Geom_ch05-01_Ex.pdf