1. **State the problem:** We have a right triangular prism with side lengths 5 in, 12 in, and 9 in. We want to find: (a) The length $a$, which is the diagonal across the triangular face. (b) The length $b$, which is the space diagonal (tape length) from one corner to the opposite corner of the prism. 2. **Formula and rules:** - For the triangular face, use the Pythagorean theorem to find the diagonal $a$: $$a = \sqrt{5^2 + 12^2}$$ - For the space diagonal $b$ of the prism, use the 3D Pythagorean theorem: $$b = \sqrt{a^2 + 9^2}$$ 3. **Calculate $a$:** $$a = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$$ 4. **Calculate $b$ using $a=13$:** $$b = \sqrt{13^2 + 9^2} = \sqrt{169 + 81} = \sqrt{250}$$ 5. **Simplify and round $b$:** $$b = \sqrt{250} = \sqrt{25 \times 10} = 5\sqrt{10} \approx 15.8$$ **Final answers:** (a) $a = 13$ in (b) $b \approx 15.8$ in