To solve this problem, we will carefully make use of the properties of similar triangles. Here's a step-by-step solution.
1. Identify the similarity ratio between triangles ABC and XYZ:
Triangle ABC has side lengths 10 units, 20 units, and 24 units, with the longest side being 24 units. Triangle XYZ is similar to triangle ABC, and its longest side is 60 units.
The similarity ratio [tex]\( k \)[/tex] is calculated by dividing the longest side of triangle XYZ by the longest side of triangle ABC:
[tex]\[
k = \frac{\text{longest side of triangle XYZ}}{\text{longest side of triangle ABC}} = \frac{60}{24} = 2.5
\][/tex]
2. Calculate the side lengths of triangle XYZ:
Using the similarity ratio, we can determine the side lengths of triangle XYZ:
[tex]\[
a_{\text{XYZ}} = 10 \times 2.5 = 25 \text{ units}
\][/tex]
[tex]\[
b_{\text{XYZ}} = 20 \times 2.5 = 50 \text{ units}
\][/tex]
[tex]\[
c_{\text{XYZ}} = 24 \times 2.5 = 60 \text{ units}
\][/tex]
3. Determine the perimeter of triangle XYZ:
The perimeter of triangle XYZ is the sum of its side lengths:
[tex]\[
\text{Perimeter}_{\text{XYZ}} = a_{\text{XYZ}} + b_{\text{XYZ}} + c_{\text{XYZ}} = 25 + 50 + 60 = 135 \text{ units}
\][/tex]
4. Calculate the area of triangle XYZ:
To find the area of triangle XYZ, we first need the height corresponding to the longest side (60 units).
Given that the height of triangle ABC with respect to its longest side is 8 units, we can use the similarity ratio to find the corresponding height in triangle XYZ:
[tex]\[
\text{height}_{\text{XYZ}} = \text{height}_{\text{ABC}} \times k = 8 \times 2.5 = 20 \text{ units}
\][/tex]
The area of triangle XYZ can now be calculated using the formula for the area of a triangle [tex]\( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)[/tex]:
[tex]\[
\text{Area}_{\text{XYZ}} = \frac{1}{2} \times 60 \times 20 = 600 \text{ square units}
\][/tex]
So, the perimeter of triangle XYZ is 135 units, and the area of triangle XYZ is 600 square units.
