In the last section, you investigated how a dilation is used to generate a new figure (image) from an original figure (pre-image). You also saw that the ratios of any pairs of corresponding sides from the dilation were equal. That means the lengths of corresponding sides are proportional, which is one of the attributes of similar figures.

Now that you’ve seen how dilations can be used to generate similar figures, you can use dilations to set up and solve proportions.

Consider the dilation shown below. Trapezoid JKLM is generated by dilating trapezoid ABCD.

Drag the tiles to set up a proportion that can be used to solve this problem by using the scale factor.

• How can you identify a pair of corresponding sides?

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  One way to identify a pair of corresponding sides is to rotate the dilated image so that it has the same orientation as the original preimage. Once the two figures are oriented the same, then you can match the sides that have the same orientation.

• Once you have identified two pairs of corresponding sides, how can you use that information to write a proportion?

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  A proportion is a statement that two ratios are equivalent. One way is to write the ratio of the lengths of two corresponding sides. To do this, the side lengths in the numerator of both ratios should be from the same figure. The denominator would be the lengths of the sides that correspond to those of the numerator.

Rotate the image so that it is oriented with the preimage. Then, match the sides of the image with the sides of the preimage.

Example 1: Solve the proportion for x.

• Find the scale factor. In this example the scale factor is 4 (3 × 4 = 12).
• Multiply by the scale factor.

3(4) over 8(4) 3(4) 8(4) = 12 over x 12 x

• Since x is in the denominator, find the denominator. x = 8(4) = 32
                                                                                     x = 32

Example 2: Solve the proportion for x.

• Find the scale factor. In this example the scale factor is 3 (7 × 3 = 21).
• Multiply by the scale factor.

2(3) over 7(3) 2(3) 7(3) = 12 over x x 21

• Since x is in the numerator, find the numerator. x = 2(3) = 6
                                                                                  x = 6

Pause and Reflect

Once you have the proportion written, how could you solve the proportion?

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Check Your Answer

One way to solve a proportion is to use a scale factor. This method is like generating equivalent fractions, only using ratios.

Practice

Use the figure below to complete the proportions that follow. Quadrilateral MNOP is a dilation of quadrilateral ABCD.

1. AD over MP AD MP = ? over AB ? ON

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   Identify the angle congruence marks for ∠O and ∠N in quadrilateral MNOP.

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   AD over MP AD MP = CB over AB CB ON

2. AB over ? AB ? = DC over PO DC PO

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   Identify the angle congruence marks for ∠A and ∠B in quadrilateral ABCD.

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   AB over MN AB MN = DC over PO DC PO

3. ? over AB ? AB = NO over BC NO BC

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   Identify the angle congruence marks for ∠A and ∠B in quadrilateral ABCD.

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   MN over AB MN AB = NO over BC NO BC