Let's break down the problem and provide a detailed, step-by-step solution:
1. Determine the scale factor between triangles [tex]\( \triangle ABC \)[/tex] and [tex]\( \triangle XYZ \)[/tex].
- Given side lengths of [tex]\( \triangle ABC \)[/tex] are 10 units, 20 units, and 24 units (where 24 units is the longest side).
- The longest side of [tex]\( \triangle XYZ \)[/tex] is 60 units.
- The scale factor is calculated by dividing the longest side of [tex]\( \triangle XYZ \)[/tex] by the longest side of [tex]\( \triangle ABC \)[/tex]:
[tex]\[
\text{Scale factor} = \frac{60}{24} = 2.5
\][/tex]
2. Calculate the perimeter of [tex]\( \triangle ABC \)[/tex].
- Sum of all side lengths of [tex]\( \triangle ABC \)[/tex]:
[tex]\[
\text{Perimeter of } \triangle ABC = 10 + 20 + 24 = 54 \text{ units}
\][/tex]
3. Calculate the perimeter of [tex]\( \triangle XYZ \)[/tex] using the scale factor.
- The perimeter of similar triangles is proportional to the scale factor:
[tex]\[
\text{Perimeter of } \triangle XYZ = 54 \times 2.5 = 135 \text{ units}
\][/tex]
4. Calculate the area of [tex]\( \triangle ABC \)[/tex].
- Given the height of [tex]\( \triangle ABC \)[/tex] with respect to its longest side (24 units) is 8 units.
- The area of [tex]\( \triangle ABC \)[/tex] can be found using the formula for the area of a triangle:
[tex]\[
\text{Area of } \triangle ABC = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 24 \times 8 = 96 \text{ square units}
\][/tex]
5. Calculate the area of [tex]\( \triangle XYZ \)[/tex] using the scale factor.
- The area of similar triangles is proportional to the square of the scale factor:
[tex]\[
\text{Area of } \triangle XYZ = 96 \times (2.5)^2 = 96 \times 6.25 = 600 \text{ square units}
\][/tex]
Thus, the correct answers are:
- The perimeter of [tex]\( \triangle XYZ \)[/tex] is 135 units.
- The area of [tex]\( \triangle XYZ \)[/tex] is 600 square units.
Combining the results back into the question format:
The perimeter of [tex]\( \triangle XYZ \)[/tex] is [tex]\( \boxed{135} \)[/tex] units. If the height of [tex]\( \triangle ABC \)[/tex], with respect to its longest side being the base, is 8 units, the area of [tex]\( \triangle XYZ \)[/tex] is [tex]\( \boxed{600} \)[/tex] square units.
