To determine the length of the hypotenuse of [tex]\(\triangle XYZ\)[/tex], given that it is similar to [tex]\(\triangle ABC\)[/tex], we can use the properties of similar triangles.
1. Step 1: Calculate the hypotenuse of [tex]\(\triangle ABC\)[/tex]
[tex]\(\triangle ABC\)[/tex] has legs of lengths 9 ft and 12 ft. To find the hypotenuse, we use the Pythagorean theorem:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
Plugging in the values for the legs:
[tex]\[ 9^2 + 12^2 = c^2 \][/tex]
[tex]\[ 81 + 144 = c^2 \][/tex]
[tex]\[ 225 = c^2 \][/tex]
[tex]\[ c = \sqrt{225} = 15 \, \text{ft} \][/tex]
So, the hypotenuse of [tex]\(\triangle ABC\)[/tex] is 15 ft.
2. Step 2: Determine the ratio of the sides of similar triangles
Since [tex]\(\triangle ABC \sim \triangle XYZ\)[/tex], the ratios of corresponding sides are equal. The shortest side of [tex]\(\triangle ABC\)[/tex] is 9 ft, and the shortest side of [tex]\(\triangle XYZ\)[/tex] is 13.5 ft. Thus, the ratio of the sides is:
[tex]\[ \frac{13.5}{9} = 1.5 \][/tex]
3. Step 3: Calculate the hypotenuse of [tex]\(\triangle XYZ\)[/tex] using the ratio
Given that the triangles are similar, the ratio of the hypotenuse of [tex]\(\triangle XYZ\)[/tex] to the hypotenuse of [tex]\(\triangle ABC\)[/tex] should also be 1.5. Therefore, we multiply the hypotenuse of [tex]\(\triangle ABC\)[/tex] by this ratio:
[tex]\[ \text{Hypotenuse of } \triangle XYZ = 15 \times 1.5 = 22.5 \, \text{ft} \][/tex]
4. Step 4: Select the best answer from the choices provided
From the given options,
A. 15 ft
B. 18 ft
C. 22.5 ft
D. 27.5 ft
The correct answer is:
C. 22.5 ft
So the hypotenuse of [tex]\(\triangle XYZ\)[/tex] is 22.5 ft.
