The table gives the perimeters of similar triangles and , where corresponds to . The length of is 18. | | Perimeter | |----------------|-----------| | Triangle | 37 | | Triangle | 333 | What is the length of ?

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SAT Area & Volume Question 2 | Free SAT Prep | Aniko

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Question 2·Medium·Area and Volume

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The table gives the perimeters of similar triangles $TUV$ and $XYZ$ , where $\overline{TU}$ corresponds to $\overline{XY}$ . The length of $\overline{TU}$ is 18.

	Perimeter
Triangle $TUV$	37
Triangle $XYZ$	333

What is the length of $\overline{XY}$ ?

For similar-figure questions involving perimeters or areas, first identify which figure is larger, then compute the ratio of the given global measures (like perimeters) to get the scale factor. Apply this scale factor directly to any corresponding side length instead of trying to set up multiple proportions, and always check that your final side length makes sense (larger figure → longer corresponding side, smaller figure → shorter corresponding side).

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Connect perimeters and side lengths

How are the perimeters of similar triangles related to the lengths of their corresponding sides?

Find the scale factor

Compute the ratio of the larger triangle's perimeter to the smaller triangle's perimeter. This ratio is the scale factor between the triangles.

Apply the scale factor to the given side

Once you know how many times larger triangle $XYZ$ is compared with triangle $TUV$ , multiply the given length $TU = 18$ by that scale factor to get $XY$ .

Desmos Guide

Compute the scale factor and side length

In Desmos, type 18 * (333/37) and press Enter. The output is the value of $XY$ , since $\dfrac{333}{37}$ is the scale factor from triangle $TUV$ to triangle $XYZ$ .

Step-by-step Explanation

Use similarity to relate perimeters and sides

For similar triangles, all linear measurements (like side lengths and perimeters) are multiplied by the same scale factor.

That means the ratio of the perimeters of triangles $XYZ$ and $TUV$ is the same as the ratio of any pair of corresponding sides, such as $XY$ and $TU$ :

\frac{\text{Perimeter of } XYZ}{\text{Perimeter of } TUV} = \frac{XY}{TU}.

Find the scale factor between the triangles

Substitute the given perimeter values into the ratio:

\frac{\text{Perimeter of } XYZ}{\text{Perimeter of } TUV} = \frac{333}{37}.

Now simplify $\frac{333}{37}$ :

$37 \times 9 = 333$ , so

\frac{333}{37} = 9.

This means triangle $XYZ$ is 9 times larger (in linear dimensions) than triangle $TUV$ , so

\frac{XY}{TU} = 9.

Use the scale factor to find the missing side

We are told $TU = 18$ and $\dfrac{XY}{TU} = 9$ , so

XY = 9 \times TU = 9 \times 18.

Compute this product:

9 \times 18 = 162.

Therefore, the length of $\overline{XY}$ is 162.

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Desmos Guide

Step-by-step Explanation

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Step-by-step Explanation