7-6

Proportional Lengths

2

, 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

3

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

4

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

5

, , , .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.

6

Angle Bisector 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

7

(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 ~ VDEF

8

A

B

C

D

E

F20 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

25

4