Perimeter and Area of Similar Quadrilaterals In this unit, you recognized how the perimeter and the area of similar triangles compare with one another. Let’s consider whether these ideas apply to quadrilaterals—shapes that have four sides.
Question 1
First you’ll study the perimeters of similar quadrilaterals. Open perimeter of similar quadrilaterals, and complete each step below.

Part A
Dilate quadrilateral ABCD with the origin as the center of dilation and a scale factor of 2.5. Record the side lengths of the corresponding sides in each quadrilateral. Round your answers to the hundredths place. Verify that the ratio of each side length of A′B′C′D′ to the corresponding side length of ABCD is equal to the dilation factor.

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Part B
Calculate the perimeter of each quadrilateral and the ratio of the perimeters. Round your answers to the hundredths place.

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Part C
Dilate quadrilateral ABCD again with the origin as the center of dilation and a scale factor of -0.5. Note the side lengths of the corresponding sides in each quadrilateral. Round your answers to the hundredths place. Verify that the ratio of each side length of A′1B′1C′1D′1 to the corresponding side length of ABCD is equal to the dilation factor.

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Part D
Calculate the perimeter of the new quadrilateral and calculate the ratio of the perimeter of the new quadrilateral with the perimeter of quadrilateral ABCD. Round your answers to the hundredths place.

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Part E
The absolute value of the dilation factor is the ratio of each side length of the dilated quadrilateral to the corresponding side length of the preimage. How does the ratio of the perimeters in parts B and D compare with the ratio of corresponding side lengths?

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Question 2
Now you will study the areas of similar quadrilaterals. Open area of similar quadrilaterals, and complete each step below.

Part A
Dilate quadrilateral ABCD with the origin as the center of dilation and a scale factor of 2. Find the areas of the preimage and the image, and calculate the ratio of the areas.

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Part B
Find the ratio of each side length of the dilated quadrilateral to the corresponding side length of the original quadrilateral. Round your answers to the hundredths place.

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Part C
How does the ratio of corresponding side lengths of the two quadrilaterals compare with the ratio of the areas of the quadrilaterals?

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Part D
Dilate quadrilateral ABCD again using the origin as the center of dilation with a scale factor of -0.5. Find the area of the new quadrilateral, and calculate the ratio of the area of new quadrilateral to the area of the original quadrilateral.

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Part E
Find the ratio of the side lengths of the new quadrilateral to the corresponding side lengths of the original quadrilateral. Round your answers to the hundredths place. Verify that they are equal to the absolute value of the dilation factor.

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Part F
How do the ratios of the lengths of corresponding sides of the quadrilaterals compare with the ratio of the areas of the quadrilaterals?

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Part G
What is the relationship between two similar quadrilaterals and their areas?

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Question 3
How do these results about area and perimeter of similar quadrilaterals compare with similar situations for triangles?

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