The ladder is the hypotenuse, and the wall height and ground distance are the legs. Use $a^2 + b^2 = c^2$ with $c = 13$ and a leg $a = 5$. Then $b = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$ feet. This models a common construction setup for checking heights safely.

The diagonal is the hypotenuse of a right triangle with legs 60 and 80. $d = \sqrt{60^2 + 80^2} = \sqrt{3600 + 6400} = \sqrt{10000} = 100$ meters. This is useful in surveying and planning straight paths across rectangular lots.

Use the converse of the Pythagorean Theorem. Compute $8^2 + 15^2 = 64 + 225 = 289$ and $17^2 = 289$. Since $8^2 + 15^2 = 17^2$, the triangle is right. Builders use this idea (like a 8–15–17 or 3–4–5 triangle) to check square corners.

Check the converse: compute $10^2 + 12^2 = 100 + 144 = 244$ and $15^2 = 225$. Because $244 \ne 225$, $a^2 + b^2 \ne c^2$, so it is not a right triangle. This kind of check helps verify whether a corner is square in design and construction.

The path forms a right triangle with legs 9 and 12. The straight-line distance is $d = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15$ kilometers. This models navigation using straight-line distance (displacement).