Question

1. a triangular prism has a right - triangular base with legs of 9 m and 12 m. if the height of the prism is 15 m, what is the space diagonal?

a. 21.21 m
b. 22.36 m
c. 24.85 m
d. 20 m

Explanation:

Step1: Find base - diagonal length

First, find the length of the diagonal of the right - triangular base using the Pythagorean theorem. For a right triangle with legs \(a = 9\)m and \(b = 12\)m, the length of the hypotenuse of the base \(d_{base}=\sqrt{a^{2}+b^{2}}\). So, \(d_{base}=\sqrt{9^{2}+12^{2}}=\sqrt{81 + 144}=\sqrt{225}=15\)m.

Step2: Find space - diagonal length

Then, consider a right - triangle formed by the base - diagonal and the height of the prism to find the space diagonal. Let the height of the prism \(h = 15\)m and the base - diagonal \(d_{base}=15\)m. Using the Pythagorean theorem again, the space diagonal \(d=\sqrt{d_{base}^{2}+h^{2}}\). So, \(d=\sqrt{15^{2}+15^{2}}=\sqrt{225+225}=\sqrt{450}\approx21.21\)m.

Answer:

A. 21.21 m