Midsegments of Midsegments of Triangles and TrapezoidsTriangles and Trapezoids

Theorems, Postulates, & Definitions

Midsegment of a Triangle: A midsegment of a triangle is a segment whose endpoints are the midpoints of two sides.

Midsegment of a Trapezoid: A midsegment of a trapezoid is a segment whose endpoints are the midpoints of the nonparallel sides.

Midsegment TheoremMidsegment Theorem

The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as the sum of the length of the top and the bottom.

( )+ B

The length of the support ST is 23 inches.

∆ Midsegment Thm.

Substitute 46 for PQ.

Simplify.ST = 23

In an A-frame support, the distance PQ is 46 inches. What is the length of the support ST if S and T are at the midpoints of the sides?

( + 0)

The diagram shows an illustration of a roof truss, where UV and VW are midsegments of RST. Find UV and RS.

UV = ½(RT + S)

UV = ½(90 + 0)

UV = 45

45 in.

57 = ½(SR + 0)

VW = ½(SR + T)

57 = ½SR

114 = SR

114 in.

Find the value ofFind the value of n. n.

n + 14 = ½(3n + 12 + 0)

n + 14 = ½(3n + 12)

2(n + 14) = 2(½(3n + 12))

2n + 28 = 3n + 12

-1n = -16n = 16

3060

MidsegmentMidsegmentA midsegmentmidsegment of a triangle is a segment that

connects the midpoints of two sides of the triangle. Every triangle has 3 midsegments.

MidsegmentsMidsegments The midsegments of a triangle divide the triangle into 4 congruent triangles

In ∆XYZ, M, N, and P are midpoints.

The perimeter of ∆ MNP is 60. Find NP and YZ.

NP + MN + MP = 60 Definition of perimeter

Because the perimeter of MNP is 60, you can find NP.

Use the Triangle Midsegment Theorem to find YZ.

NP + 24 + 22 = 60 Substitute 24 for MN and 22 for MP.NP + 46 = 60 Simplify.

MP = (YZ + X) Triangle Midsegment Theorem12

22 = (YZ + 0) Substitute 22 for MP.12

44 = YZ Multiply each side by 2.

NP = 14 Subtract 46 from each side.

14

1. ED

2. AB

3. mBFE

10

14

Find each measure.Find each measure.

44° Corresponding Angles

∆XYZ is the midsegment triangle of

∆WUV. What is the perimeter of ∆XYZ?

4.5 + 4 + 3 = 11.5

1. XY

2. VW

3. XZ

4. Perimeter

8

4

44.5

4.5

Cases with more than one Parallel LineCases with more than one Parallel Line

Difference of the Bases divided by the number of spaces.

40

40

60 – 0 = 60

0

60

Difference of Bases20

20

Number of Spaces

30 – 10 = 20

15

15

20

20

25

25

16x22

7x55x3

1

16x2

2

12x8

8x + 12 = 2(2x + 16)

8x + 12 = 4x + 32

4x = 20

x = 5

EF = 26

a.a. b.b. c.c.x = 9x = 9 x = 14x = 14 x = 11x = 11

d.d. e.e. f.f.x = 23.5x = 23.5 x = 7x = 7 x = 2x = 2

Solve For The Variable in a – f.

g.g. = 40= 40 h.h. = 50= 50

i.i. = 160= 160 j.j. = 80= 80

Solve For The Variable

x.x. y.y.x = 6x = 6 y = 6.5y = 6.5

Assignment

3.7A and 3.7B3.7A and 3.7BSection 10 - 16Section 10 - 16