Download Proportional Parts Lesson 5-4

Lesson 5-4
Proportional
Parts
Lesson 5-4: Proportional Parts
1
Similar Polygons
Two polygons are similar if and only if their corresponding angles
are congruent and the measures of their corresponding sides are
proportional.
AB AC BC


DE DF EF
F
C
A
B
D
E
Lesson 5-4: Proportional Parts
2
Triangle Proportionality Theorem
If a line is parallel to one side of a triangle and intersects the other
two sides in two distinct points, then it separates these sides into
C
segments of proportional length.
If BD AE , then
CB CD

BA DE
B 1
Converse:
If a line intersects two sides of a triangle and
A 4
separates the sides into corresponding segments
of proportional lengths, then the line is parallel
to the third side.
CB CD
If
BA

DE
Lesson 5-4: Proportional Parts
2 D
3 E
, then BD AE
3
Examples………
Example 1: If BE = 6, EA = 4, and BD = 9, find DC.
6 9
9

4 x
D
6x = 36
x
x=6
C
Example 2: Solve for x.
2x + 3
E
5
A
B
6
E
4
A
2x  3 4x  3

5
9
5(4 x  3)  9(2 x  3)
B
4x + 3
20 x  15  18 x  27
D
9
2 x  12
C
Lesson 5-4: Proportional Parts
x6
4
Theorem
A segment that joins the midpoints of two sides of a triangle is
parallel (P) to the third side of the triangle, and its length is onehalf the length of the third side.
R
If L is the midpo int of RS and
M is the midpo int of RT then
1
LM ST and ML  ST .
2
M
L
T
S
Lesson 5-4: Proportional Parts
5
Corollary
If three or more parallel lines have two transversals, 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
E
transversal.
D
AB DE AC BC AC DF

,

,

, etc.
BC EF DF EF
BC EF
Lesson 5-4: Proportional Parts
A
B
C
6
F
Theorem
An angle bisector in a triangle separates the opposite side into
segments that have the same ratio as the other two sides.
AD AC
If CD is the bi sec tor of ACB, then

DB BC
C
A
D
Lesson 5-4: Proportional Parts
B
7
If two triangles (V) are similar:
(1) then the perimeters are proportional to the measures of the
corresponding sides.
(2) then the measures of the corresponding altitudes are proportional
to the measure of the corresponding sides..
(3) then the measures of the corresponding angle bisectors of the
triangles are proportional to the measures of the corresponding sides..
AB BC AC Perimeter of ABC



DE EF DF Perimeter of DEF
A
D
B
G
H
C
ABC ~
E
I
AG (altitude of ABC )

DI (altitude of DEF )
J
DE
F
F

AH (angle bi sec tor of ABC )
DJ (angle bi sec tor of DEF )
Lesson 5-4: Proportional Parts
8
Example:
Given: ΔABC ~ ΔDEF, AB = 15, AC = 20, BC = 25, and DF = 4.
Find the perimeter of ΔDEF.
The perimeter of ΔABC is 15 + 20 + 25 = 60.
Side DF corresponds to side AC, so we can set up a proportion as:
AC Perimeter of

DF Perimeter of
20 60

4
x
20 x  240
x  12
ABC
DEF
E
B
25
15
A
F
D
4
C
20
Lesson 5-4: Proportional Parts
9