Question

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△abc has side lengths of 10 units, 20 units, and 24 units. △xyz is similar to △abc, and the length of its longest side is 60 units.
the perimeter of △xyz is ▼ units. if the height of △abc, with respect to its longest side being the base, is 8 units, the area of △xyz is ▼ square units.
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Explanation:

Step1: Find the scale factor

The longest side of \( \triangle ABC \) is 24 units, and the longest side of \( \triangle XYZ \) is 60 units. The scale factor \( k \) is \( \frac{60}{24}=\frac{5}{2} \).

Step2: Find the perimeter of \( \triangle ABC \)

The perimeter of \( \triangle ABC \) is \( 10 + 20 + 24 = 54 \) units. Then the perimeter of \( \triangle XYZ \) is \( 54\times\frac{5}{2}=135 \) units.

Step3: Find the area of \( \triangle ABC \)

The area of \( \triangle ABC \) with base 24 and height 8 is \( \frac{1}{2}\times24\times8 = 96 \) square units.

Step4: Find the area of \( \triangle XYZ \)

For similar triangles, the ratio of areas is the square of the scale factor. The scale factor is \( \frac{5}{2} \), so the ratio of areas is \( (\frac{5}{2})^2=\frac{25}{4} \). Then the area of \( \triangle XYZ \) is \( 96\times\frac{25}{4}=600 \) square units.

Answer:

The perimeter of \( \triangle XYZ \) is 135 units. The area of \( \triangle XYZ \) is 600 square units.