Transcript 7.4 Parallel Lines and Proportional Parts (1).

Parallel Lines and
Proportional Parts
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
C
D
AB||ED, BD = 8, DC = 4, and AE = 12.
C
• Find EC
E
D
A
B
•
CD
DB
4
8
•


CE
EA
by
EC
12
EC = 6
?
UY|| VX, UV = 3, UW = 18, XW = 16.
• Find YX.
UV

WV
3
15

U
V
YX
XW
YX
W
16
• YX = 3.2
Y
X
Converse of Side Splitter
• 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 A B  A E
BC
ED
• then BE || CD
B
C
E
D
Determine if GH || FE. Justify
• In triangle DEF, DH = 18, and HE = 36,
and DG = ½ GF.
• To show GH || FE,
• Show D G  D H
GF
HE
• Let GF = x, then DG = ½ x.
• Substitute
1
x
2
x
• Simplify

36
1
2  18
1
36
• Simplify
1
2

18
1
2
• Since the sides are proportional, then GH
|| FE.
Triangle Midsegment Theorem
• 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
Example
• 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  4 
2
2 

•
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 = 6 8
which simplifies to 2 1 7
DE = 1 7
DE = ½ BC
Corollaries of side splitter thm.
• 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.
Examples
• 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 2nd corollary to say DE = EF
3y = 5/3 y + 1
Y=¾