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I^otes: Ratios and Proportions

the comparison o f two numbers using C^i'i/t'S/'o^ ,

written'^a to b", a:b, or ^ (b 7̂ 0)

Proportion an equation that states two rdho are equal; ex: 7 = (b 0, d ^ 0)

b a

Cross Products In a proportion, the product of the extremes equals the product o f the means:

I f - = then ai = L-c b d

• Ratios should be expressed in simplest form, and can be used to find missing dimensions • In a ratio, the denominator of the fraction cannot be zero because division by zero is undefined.

Example 1: F i n d the r a t i o o f the w i d t h to the l e n g t h o f the rectangle. S i m p l i f y the ra t io .

4 cm

12 cm

• Proportions are made up of parts called means and extremes:

5 4

ext remes — ^ t , ; ^ means

means —• < ^ — extremes

Example 3:

Vh-/6> - too •J / o

Example 4:

2y 8 4y

• Extended ratios compare more than two numbers.

Example 5: Find the measures of the angles of the triangle, i f the angle measures have a ratio of 1:7:10.

/ go ' Example 6: The ratio of the side lengths of a triangle is 6:7:9. and its perimeter is 99 meters. What is the length of the longest side?

scale &y>

S.lNotes: S imi lar Polygons

Congruence: Same shape and size (=)

Have the s ame 5kaf>^ b u t n o t necessa r i ly the s ame SiZe^

*xve use the symbol (~) to denote similarity

• Similar Figures: Corryesponding angles are ^ corresponding sides are

• Scale F a c t o r / S i m i l a r i t y Ratio: t he c o m m o n r a t i o o f the c o r r e s p o n d i n g s ide l e n g t h s i n

s i m i l a r f igures . " ' "

Example 1: A A B C ~ A D E F

AB

Z DE to

BC _ 4_

AC _ 5_ _ /

DF~ Cp ~ ^

The scale factor/similarity ratio is:

Example 2: Identify the pairs of congruent angles and corresponding sides. .

C = 2>

Z B = Z ^

And by third angle theorem:

Z A ^ Z P

These t r i a n g l e s are s i m i l a r because :

D F ^

EF y

DE (o

'2 3 W h a t is t he s i m i l a r i t y r a t i o of: A A B C a n d A D E F

W r i t e t h e s i m i l a r i t y s t a t e m e n t for the t w o t r i a n g l e s : A A^t/^ ~ A

Example 3: A D E F ~ AMNP . F i n d x.

N

D — T ? F M 16

V

Example 4: A B C D E - F G H J K .

F i n d the scale factor (ratio):

F i n d x =

A 10 B

5 v = "SC* '8

F i n d the perimeter of ABCDE= JOJ^^ ^>o \i'2.'=-4Lp

F i n d the perimeter of FGHJK= t^*H*-n f ^ - / S - 6>^

E 10 D

15

18

K 15

What w o u l d the scale factor (ratio) of the perimeters be? Co'? '

What re la t ionship do y o u see between the two scale factors (ratios)? •Sc3/^£^

Notes: Change in Dimension

Similarity, Perimeter and Area Ratios

Scale factor

Example 1: ^ ~^ ^

Ratio of Perimeter

a:b a

b

Ratio of Area

a'

R a t i o of C o r r e s p o n d i n g S ide lengths

R a t i o of P e r i m e t e r s R a t i o of A r e a s

4:5 7 : ^ 7:9

1^:^ 144:25

-iT/se the figures below to answer questions #1 - 9.

A

4 cm

10 cm

6 cm

1. Find the scale factor o f the sides. ( E F G H M B C ^ )

2. Find the perimeter o f \ ^ C D

3. Find the perimeter o f E F C i l l

4. Find the scale factor o f the perimeters (EFGH / AB C D)

5. How does the scale factor o f the sides compare to the scale factor o f the perimeter?

^he^^ are. M e sar^e., 6. Find the a;;ea o f A B C D ^ '

7. Find the area o f EFGH 1 \

8. Find the scale factor o f the areas (EFGH/ABCD)

9. How does the scale factor

15 cm

3f the sides com Dare to the sc'aie factor o f the area? ^

Ex. 2

Find the scale factor of AEFG ~ AHJK:

find the perimeter AHJK:

find the area of AHJK:

I %2

Ex 3

Tony and Edwin each bui l t a rectangular garden. Tony's garden is twice as long and twice as wide as Edwin's garden. If the area of Edwin's garden is 600 square feet, wha t is the area of Tony's garden?

I

C-

Notes: Methods of proving Triangles Similar

Rp'-i l l what s imilar i ty means: 1] Corresponding angles are COnQroC*^ 2] The ratios of the measures of corresponding sides are.

Angle- Angle Similarity ( A A ~ ) - If two angles of one triangle are = ^

another triangle, then the triangles are similar

. to two corresponding angles of

^ ^^^^

Side- Side- Side Similarity (SSS~) - If the three sides of one triangle are ^'^g^cx- \\. / to the t h r ^

corresponding sides of another triangle, then the triangles are similar.

In this example, the ratios of sides are: . a : x ^ 6 : 7.5 = 1 2 : 15 = 4 : 5 . b : v = 8 : 10 = 4 : 5 • c: z = 4 j j ^ ' " '

These ratios are all equal, so the two triangles are similar.

Side-Angle- Side Similarity (SAS~) - I f t w o sides of one triangle are ^rp^p/' {'•ong^ ( to two

corresponding sides of another triangle and their included angles are ^ then the triangles are similar.

In this example we can see that:

• one pa i r of s ides is in t he ra t i o of 2 1 : 14 = 3 : 2

• a n o t h e r pair o f s ides is in t he ra t i o o f 15 : 10 = 3 : 2.

• t h e r e is a ma t ch i ng ang le of 75° in be tween t h e m

*The 3 ways to prove s imi lar triangles are: , and ^^^"^

1^

Examples

Decide if each pair of triangles is s imilar. If they are, write the correspondence in the first blank and the reason in the second blank. If they are NOT similar, write NS in the second blank.

1) A A B C - A 96(^ by -^^S ^ 7 2) A A B C - A S)i(^ by Ak^ B

3) A J KL ~ A J / A / 4) ATXU-A ^/Xiy} by S J S

Example 5: Explain why the triangles are similar, then find BE and CD

+ 6

Example 6: Find the value o f x that makes AFGH ~ A J K L .

( v - j ) (y^i.^

2S

i

Notes Chapter 8.4 Properties of Similar Triangles and Proportional Relationships

Tr iangle Proport ional i ty T h e o r e m E x a m p l e

If a line parallel to a side of a triangle intersects the other two sides, then it divides those sides f^^ofori^^o^M.

So BX _ BY ^°XA-YC

ex 1: Find PN ex 2: Find US u,

1 4 , •

C o n v e r s e of the Triangle Proportionality T h e o r e m

If a line divides two sides of a trianqle,proportionally, then it is

fhrafleJtn the third side.

E x a m p l e

e x - - Q L X4 v c ^

ex 3. Verify that D E / / B C 7

8 f 12 C

IS

2̂

Tr iangle Ang le B i s e c t o r T h e o r e m E x a m p l e

An 3nglA^''Sfic>4

Two Transversal Proportionality Corollary: If three or more parallel lines intersect two transversals, then they divide the transversals proportionally.

CE BD DF

ex 4. Suppose that an artist decided to make a larger slcetch of the trees. In the figure, i f AB = 4.5 in., BC = 2.6 in.. CD = 4.1 in., and KL = 4.9 in., find L M and MN to the nearest tenth of an inch, / i ^ ,/ j

A/ore-.' V f̂ifi'S pareJl&i

2.6 in M - J ^ " ' Che.' y t-^ " x ^

Ex 5 : In Goodville Oak, Pine, Cedar and Maple Streets are parallel streets that intersect Main Street and Central Avenue, as shown in the diagram.

Main St.

Central Avenue A. How long to the nearest foot is Central Avenue from Pine Street to Cedar Street?

B. How long is Central Avenue from Oak Street to Maple Street? ^

580 r

Notes: Indirect measurement

E x . 1 A s tudent w h o is 5 ft 6 in. tall m e a s u r e d s h a d o w s to find the height LM of a f lagpole . What is L M ?

^0 _ no " X / 7 o

14 ft 2 in.

Ex 2 T h e r e c t a n g u l a r cen t ra l c h a m b e r of the L inco ln Memorial is 74 ft long a n d 60 ft w i d e . Make a s c a l e d r a w i n g of the f loor of the c h a m b e r us ing a s c a l e of 1in : 20ft.

X

scale

(,0 3 " /

10 "^^

E x . 3 In the f igure, A D B A = A E C A . What is the d i s t a n c e a c r o s s the l a k e ?

1^

E x . 4 F ind the height of the t ree .

7 ^ /2. 4

e 12 f1 C D 3 ft

E

JL -

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