lesson 5-4: proportional parts 1 proportional parts lesson 5-4
TRANSCRIPT
Lesson 5-4: Proportional Parts 1
Proportional Parts
Lesson 5-4
Lesson 5-4: Proportional Parts 2
Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional.
A B
C
D E
F
AB AC BC
DE DF EF
Similar Polygons
Lesson 5-4: Proportional Parts 3
, CB CD
If BD AE thenBA DE
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 segments of proportional length.
Triangle Proportionality Theorem
1 2
34A
B
C
D
E
Converse:If a line intersects two sides of a triangle and separates the sides into corresponding segments of proportional lengths, then the line is parallel to the third side.
, CB CD
If then BD AEBA DE
Lesson 5-4: Proportional Parts 4
6 9
4 x
4x + 3
9
A
B
C
DE
2x + 3
5
2 3 4 3
5 95(4 3) 9(2 3)
20 15 18 27
2 12
6
x x
x x
x x
x
x
A
B
C
D E
If BE = 6, EA = 4, and BD = 9, find DC.
6x = 36 x = 6
Solve for x.
Example 1:
Example 2:
Examples………
6
4
9
x
Lesson 5-4: Proportional Parts 5
Theorem
A segment that joins the midpoints of two sides of a triangle is parallel to the third side of the triangle, and its length is one-half the length of the third side.
R
S T
ML
int
int
1.
2
If L is the midpo of RS and
M is the midpo of RT then
LM ST and ML ST
Lesson 5-4: Proportional Parts 6
, , , .AB DE AC BC AC DF
etcBC EF DF EF BC EF
If three or more parallel lines have two transversals, they cut off the transversals proportionally.
AB
C
D
EF
Corollary
If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.
Lesson 5-4: Proportional Parts 7
Theorem
An angle bisector in a triangle separates the opposite side into segments that have the same ratio as the other two sides.
sec ,AD AC
If CD is the bi tor of ACB thenDB BC
C
A
BD
Lesson 5-4: Proportional Parts 8
(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..
B C
A
E F
D
HG I J
If two triangles are similar:
( )
( )
( sec )
( sec )
AG
D
Perimeter of ABC
Perimeter of DEF
altitudeof ABC
altitudeof DEF
anglebi tor of ABC
I
AH
DJ anglebi tor
AB BC AC
DE EF DF
of DEF
~ABC DEF
Lesson 5-4: Proportional Parts 9
A
B
C
D
E
F
20 60
420 240
12
AC Perimeter of ABC
DF Perimeter of DEF
xx
x
The perimeter of ΔABC is 15 + 20 + 25 = 60.Side DF corresponds to side AC, so we can set up a proportion as:
Given: ΔABC ~ ΔDEF, AB = 15, AC = 20, BC = 25, and DF = 4. Find the perimeter of ΔDEF.
Example:
15
20
254