5-1 Midsegments of Triangles (Notes)



5-1 Midsegments of Triangles (Notes)

A midsegment of a triangle is a segment connecting the midpoints of two sides.

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.

One way to prove the Triangle Midsegment Theorem is to use similar triangles. Another way is to use coordinate geometry and algebra. This style of proof is called a coordinate proof.

Given: [pic]TRI where T(0,0) R(6, 10) I(8, 0)

D is the midpoint of [pic]

F is the midpoint of [pic]

Verify: [pic] and DF = ½ RI

Sample Problems:

1. M, N, and P are midpoints; MP = 22; MN = 24; perimeter of ∆MNP = 60.

Find NP and YZ

2. Find m[pic]AMN and m[pic]ANM.

3. [pic] is a new bridge being built over a lake as shown. Find the length of the bridge.

4. The perimeter of a triangle is 78 ft. Find the perimeter of the triangle formed by its midpoints.

5. Only one of the lengths a, b, or c can be found. Name the segment and find its length.

6. DE = x + 5 and BC = 3x – 1. Find the value of x.

-----------------------

X

Z

Y

P

M

N

B

N

M

C

A

|nopqrstÑÓÔÕÖ×ØÙÚÜýýýýýýýýýýýýý75˚

D

C

963 ft

2640 ft

963 ft

4

6

5

4

3

3

c

b

a

17

17

14

14

A

E

D

C

B

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