4.4

How Can I Use Equivalent Ratios?

Pg. 13Applications and Notation

4.4–How Can I Use Equivalent Ratios?Applications and Notation

Now that you have a good understanding of how to determine similarity, you are going to use proportions to find missing parts of similar shapes.

4.19 – EQUAL RATIOS OF SIMILARITYCasey wants to learn more about her enlarged "C's".  a. Since the zoom factor multiplies each side of the original shape, then the ratio of the widths must equal the ratio of the lengths.  Casey decided to show these ratios in the diagram at right. Verify that her ratios are equal by reducing each one.

24

8

18

6

3

1

3

1

8

6

4=3

24

18

4=3

c. She decided to create an enlarged "C" for the door of her bedroom. To fit, it needs to be 20 units tall. If x is the width of this "C", write and solve an proportion to find out how wide the "C" on Casey's door must be. Be ready to share your equation and solution with the class.

8=

68

20=

8x = 120x = 15

8x = 120x = 15

20x

6

x

4.20 – PROPORTIONSUse your observations about ratios between similar figures to answer the following: a. Are the triangles similar? How do you know?

6=

32

1

10=

52

1

7

4Not similar

b. If the pentagons at right are similar, what are the values of x and y?

8=

24

24y = 144

y = 6

y18

8=

24

8x = 264x = 33

11x

4.21 – PROPORTIONS Your team may have used a proportion equation to solve for the previous problem. It is important that parts be labeled to help you follow your work. The same measures need to match to make sure you will get the right answer.

Likewise, when working with geometric shapes such as the similar triangles below, it is easier to explain which sides you are comparing by using notation that everyone understands.

AC DF=

CB FE

AB DE=

BC EF

AB AC=

DE DF

Yes, angles need to add to 180°

a. What other angles should match up?

B XC Y

b. Complete the similarity statement for the triangles.

ABC ~ ___________ZXY

Examine the triangles below. Which of the following statements are correctly written and which are not? Hint: more than one statement is correct.

4.23 – READING SIMILARITY STATEMENTS Read the similarity statements below. Determine which angles must be equal. Then determine which sides match up.

D

E

F

DE

EF

DF

F

U

N

FU

UN

FN

4.24 – PROPORTION PRACTICE Find the value of the variable in each pair of similar figures below. Make sure you match the correct sides together.

a. ABCD ~ JKLM

12=

x6

96x = 108x = 18

NOP ~ XYZ

5=

w3

123w = 60

w = 20

GHI ~ PQR

7=

163n

7n = 48n = 6.86

ABC ~ XYZ

7=

m

10

11

10m = 77m = 7.7

4.25 – NESTING TRIANGLES Rhonda was given the diagram and told that the two triangles are similar. a. Rhonda knows that to be similar, all corresponding angles must be equal. Are all three sets of angles equal? How can you tell?

AEB ADC ABE ACD A A

 b. Rhonda decides to redraw the shape as two separate triangles, as shown. Write a proportional equation using the corresponding sides, and solve. How long is AB? How long is AC? 4

4 7 8

x

x

4x + 32 = 11x

32 = 7x

4.57 = x