Section 7.4

Proportional parts of triangles
•
Non parallel transversals that intersect parallel lines can be extended to form similar triangles.
•
So, the sides of the triangles are proportional.

Side Splitter Theorem or Triangle
Proportionality Theorem
•
•
If a line is parallel to one side of a triangle and intersects the other two sides, it divides those two sides proportionally.
A
•
If BE || CD, then
B E AB

AE
BC ED
D C

AB||ED, BD = 8, DC = 4, and AE = 12.
C
•
Find EC
D E
•
•
CD

CE
DB EA
4

EC
8 12 by
EC = 6
B
?
A

UY|| VX, UV = 3, UW = 18, XW = 16.
•
Find YX.
UV

YX
WV XW
3

YX
15 16
•
YX = 3.2
Y
U
X
V
W

•
If a line intersects the other two sides and separates the sides into corresponding segments of proportional lengths, then the line is parallel to the third side.
A
•
If
BC ED
B E
• then BE || CD
C D

•
In triangle DEF, DH = 18, and HE = 36, and DG = ½ GF.
•
To show GH || FE,
•
Show DG

DH
GF HE
• Let GF = x, then DG = ½ x.

• Substitute
1
2 x x

18
36
•
Simplify 1
2 
18
1 36
• Simplify 1

1
2 2

•
Since the sides are proportional, then GH
|| FE.

•
A midsegment of a triangle is parallel to one side of the triangle, and its length is one-half the length of that side.
•
If D and E are mid-
•
Points of AB and AC,
•
Then DE || BC and
• DE = ½ BC

•
•
•
•
•
Triangle ABC has vertices A(-2,2), B(2, 4) and C(4,-4). DE is the midsegment of triangle ABC.
Find the coordinates of D and E.
D midpt of AB


 
2
2 2
,

2
4


D(0,3)

•
E midpt of AC
•
E(1, -1)
•
Part 2 - Verify BC || DE
•
Do this by finding slopes
•
Slope of BC = -4 and slope of DE = -4
•
BC || DE

•
Part 3 – Verify DE = ½ BC
•
To do this use the distance formula
•
BC = which simplifies to
2 17
•
DE = 17
• DE = ½ BC

•
1. If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally.

•
If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

•
1. In the figure, Larch, Maple, and
Nutthatch Streets are all parallel. The figure shows the distance between city blocks. Find x.

•
Find x and y.
•
Given: AB = BC
•
3x + 4 = 6 – 2x
•
X = 2
•
Use the 2 nd corollary to say DE = EF
•
3y = 5/3 y + 1
• Y = ¾