Geometry

7.6 Proportional Lengths

Proportional Lengths

AC and XZ are divided proportionally if…

. . .

. . .X

A B C

Y Z

=BC

XYAB

YZ

Example:

2 9

184=

2

9 18

4

Triangle Proportionality Thm. If a line parallel to one side of a triangle intersects

the other two sides, then it divides those sides

proportionally. big A

small A

whole Abig B

small B

whole B=

side C1

side C2

=

=

= = = =

whole A

small A

whole B

small B

side C1

side C2

big A

small A

big B

small B

big A

whole A

big B

whole B

whole Bwhole A

big A big B

small A

small B

big A

big B

whole A

whole B

All of these proportions, and their inverses, work.The key is to use the easiest one to solve each problem.

Think of it as two separate similar triangles.

Corollary

If three // lines intersect two transversals,…then they divide the transversals

proportionally.

a

b

c

d =ab

cd

Triangle Angle Bisector Thm. If a ray bisects an angle of a triangle,… then it divides the opposite side into

segments proportional to the other two sides.a

bc

d

=ab

cd

Directions: Use the given information and the triangle above to find the missing segment.

3. GE = 15 DE = 27 HF = 20 DH = ?

4. DE = 20 HF = 4 DH = 12 DG = ?

E F

G H

D

Directions: Use the given information and the diagram to find x.

2. AB = 27 BC = 18 DE = x + 10 EF = x

3. AB = 25 – x BC = x DE = 16 EF = 4

A

B

C

D

E

F

Directions: is and angle bisector of . Find x.

2. RP = x PQ = 12.5 RS = 8 SQ = 10

3. RP = 10 PQ = 20 RS = x RQ = 15

P

QR S

PS .RPQ

HW

P. 271 (1-7) P. 272-273 (1-14, 20, 21) Quiz 7.4-7.6 Tomorrow

A few from the HW

P. 272 #5, #14

An Example

15

520

12

416

=

18 24

=

=

= =

= =

20

5

16

4

18

24

15

5

12

4

15

20

12

161620

15 12

5

4

15

12

20

16

Solve for x. (figure not to scale)

15

27

12

20

x

= x

15

12

20

= x

5

4

20

Reduce by 3

times 4 equals

times 4 equals

x = 16

Solve for x.

18

27

x

x + 10

=18

27

x

x + 10

Reduce by 9.

=2

3

x

x + 10

2(x + 10) = 3x

2x + 20 = 3x

x = 20

20

30

Solve for x.

1512

18

x

=12

15

x

18 - x

18 - x

Reduce by 3.

=4

5

x

18 - x

4(18 – x) = 5x

72 – 4x = 5x

9x = 72

x = 8

8 10