Proportional Proportional Lengths of a Lengths of a TriangleTriangleKeystone Geometry

Remember this Remember this Theorem?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. This is a midsegment.

R

S T

ML

If L is the midpo int of RS and

M is the midpo int of RT then

LM PST and ML =12

ST

3

, =PCB 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.

This also works for Proportions: Triangle

Proportionality Theorem

1 2

34A

B

C

D

EConverse: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. ,= P

CB CDIf then BD AE

BA DE

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

5

, , , .AB DE AC BC AC DF

etcBC EF DF EF BC EF

= = =

This also works for proportions: If three or more parallel lines have two transversals, they cut off the transversals proportionally.

AB

C

D

EF

Remember this Corollary? If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

Example: AB is parallel to CD and CD is parallel to EF. Solve for x, AC, and CE.

6

9=x+5

3x18x=9(x+5)

18x=9x+45

9x=45

x=5

7

Angle Bisector Angle Bisector TheoremTheorem

Definition: An angle bisector is a line segment that bisects one of the vertex angles of a triangle. In a triangle, the angle bisector separates the opposite side into segments that have the same ratio as the other two sides.

If CD is the bisec tor of ∠ACB,

thenADDB

=ACBC

C

A

BD

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 If two triangles are similar: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

= = =

=

=

VV

VV

VV

VABC ~ VDEF

9

A

B

C

D

E

F

20 60

420 240

12

AC Perimeter of ABC

DF Perimeter of DEF

xx

x

=

=

==

VV

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