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C

A

B

D

E

Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional.

Similar Polygons

Lesson 5-4: Proportional Parts

B

1

2

D

A

4

3

E

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 segments of proportional length.

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.

Lesson 5-4: Proportional Parts

D

E

6

9

C

A

x

4

B

4x + 3

2x + 3

D

E

9

5

A

C

Examples………

Example 1:

If BE = 6, EA = 4, and BD = 9, find DC.

6x = 36 x = 6

Example 2:

Solve for x.

Lesson 5-4: Proportional Parts

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

M

L

T

S

Lesson 5-4: Proportional Parts

If two triangles are similar:

(1) then the perimeters are proportional to the measures of the corresponding sides.

Lesson 5-4: Proportional Parts

B

F

D

C

A

25

15

4

20

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:

Lesson 5-4: Proportional Parts

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