Question

select the correct answer from each drop - down menu.
δabc has side lengths of 10 units, 20 units, and 24 units. δxyz is similar to δabc, and the length of its longest side is 60 units.
the perimeter of δxyz is units. if the height of δabc, with respect to its side being the base, is 8 units, the area of δxyz is square units.

Explanation:

Step1: Calculate the perimeter of \(\triangle ABC\)

The perimeter of \(\triangle ABC\) is \(P_{ABC}=10 + 20+24=54\) units.

Step2: Find the scale factor

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

Step3: Calculate the perimeter of \(\triangle XYZ\)

Using the property that if two similar triangles have a scale factor \(k\), then the ratio of their perimeters is also \(k\). Let \(P_{XYZ}\) be the perimeter of \(\triangle XYZ\). Then \(P_{XYZ}=k\times P_{ABC}\)
\(P_{XYZ}=\frac{5}{2}\times54 = 135\) (This part seems wrong in the original dropdown options, maybe there is a mis - understanding. Let's re - check. Wait, no, maybe the first blank is wrong in the user's problem setup. Let's calculate the area part correctly)

Step4: Calculate the area of \(\triangle ABC\)

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

Step5: Calculate the area of \(\triangle XYZ\)

Since the ratio of the areas of two similar triangles is \(k^{2}\) (where \(k\) is the scale factor). \(k=\frac{60}{24}=\frac{5}{2}\), and \(A_{XYZ}=k^{2}\times A_{ABC}\)
\(A_{XYZ}=(\frac{5}{2})^{2}\times96=\frac{25}{4}\times96 = 600\) square units

Answer:

The perimeter of \(\triangle XYZ\) is \(135\) units (but if we assume there is a mistake in the problem's perimeter dropdown and focus on the area part which is correctly calculated as \(600\) square units)