Triangles and Lines - Proportional Relationships A proportion is just a comparison of two ratios or fractions. They are equivalent fractions. As one dimension

$Page 1: Triangles and Lines - Proportional Relationships A proportion is just a comparison of two ratios or fractions. They are equivalent fractions. As one dimension$
Triangles and Lines - Proportional Relationships

A proportion is just a comparison of two ratios or fractions. They are equivalent fractions. As one dimension increases or decreases by a factor, the other dimension increases or decreases by the same factor.

It is widely used in blue prints when scaling drawings. You can put a ten story building on a 8.5 x 11 sheet of paper. Where one inch on the paper represents ten feet in actual size.




d

c

b

a

Here is a typical proportion…




d

c

b

a


You solve a proportion by cross multiplying…

bcad




d

c

b

a



bcad

15

10

3

2

3030

103152




d

c

b

a



bcad

15

10

3

2

3030

103152

You try one…

123

1 x




d

c

b

a



bcad

15

10

3

2

3030

103152

You try one…

123

1 x

x

x

x

4

312

3121


The proportionality theorem describes relationship of parallel lines that go thru triangles. The line is parallel with one of the sides of the triangle.

A

ED

C B



A

ED

C B

Theorem : If a line is parallel with one side of a triangle and intersects the other two sides, then it divides those sides proportionally.



A

ED

C B

Theorem : If a line is parallel with one side of a triangle and intersects the other two sides, then it divides those sides proportionally.

So the measure of AD and DC is proportional to the measure of AE and EB.

EB

AE

DC

AD



A

ED

C B

This works because DE creates two similar triangles; ∆ADE and ∆ABC.

A



A

ED

C B

A


Similar Triangles have equal angles and proportional sides.



A

ED

C B

A



So if AC = 8 and AD = 4, we have a factor of 2. ∆ABC’s sides are all two times larger than ∆ADE’s sides.

8

4



A

ED

C B

A




What would AE equal if AB = 14 ?

8

4

14



A

ED

C B

A




What would AE equal if AB = 14 ? AE = 7

8

4

14

7


Let’s solve some problems…

A

E

D

C B

Example # 1 : Find the measure of AD if AB ║ DE.

x

6

4

9



A

E

D

C B

EBDCCEAD


x

6

4

9

You don’t really have to memorize the rule, just multiply ACROSS the parallel line…



A

E

D

C B

6

366

946

x

x

x

EBDCCEAD


x

6

4

9

You don’t really have to memorize the rule, just multiply ACROSS the parallel line…



A

ED

C B

Example # 2 : Find the measure of EB if CB ║ DE.

x

16

9

11



A

ED

C B

Example # 2 : Find the measure of EB if CB ║ DE.

x

16

9

11

x

x

x

09.13

11144

11169



A

E

D

C B

Example # 3 : Is AB ║ DE ?.

8 3

6

2



A

E

D

C B

Example # 3 : Is AB ║ DE ?. NO !!!

8 3

6

2

1816

6382


Parallel Proportional Segments theorem - if three or more parallel line are cut by two transversals, intercepted

segments on the transversals are proportional.




a

b

c

- Lines a, b, and c are parallel




a

b

c

-Lines a, b, and c are parallel

-Transversals t1 and t2 cut lines a, b, and c

t1 t2

A

B

C

D

E

F




a

b

c

-Lines a, b, and c are parallel

-Transversals t1 and t2 cut lines a, b, and c

t1 t2

A

B

C

D

E

F

DEBCEFAB




a

b

c

Example # 1 : Find the measure of segment AB.

t1 t2

A

B

C

D

E

F

DEBCEFAB

10 12

9x




a

b

c

Example # 1 : Find the measure of segment AB.

t1 t2

A

B

C

D

E

F

DEBCEFAB

10 12

9

x

x

x

5.7

1290

12910

Again, just multiply across the parallel line…

x




a

b

c

Example # 1 : Are lines a, b, and c parallel ?.

t1 t2

A

B

C

D

E

F

DEBCEFAB

10 20

126




a

b

c

Example # 1 : Are lines a, b, and c parallel ?.

t1 t2

A

B

C

D

E

F

DEBCEFAB

10 20

126

120120

1210206

YES !!!

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Triangles and Lines - Proportional Relationships A proportion is just a comparison of two ratios or fractions. They are equivalent fractions. As one dimension