Question

Two similar right triangles have areas of 6 square inches and 150 square inches. The length of the hypotenuse of the smaller triangle is 5 inches. What is the sum of the lengths of the legs of the larger triangle?

Answer 1

$\bf \qquad \qquad \textit{ratio relations} \\\\ \begin{array}{ccccllll} &Sides&Area&Volume\\ &-----&-----&-----\\ \cfrac{\textit{similar shape}}{\textit{similar shape}}&\cfrac{s}{s}&\cfrac{s^2}{s^2}&\cfrac{s^3}{s^3} \end{array} \\\\ -----------------------------\\\\ \cfrac{\textit{similar shape}}{\textit{similar shape}}\qquad \cfrac{s}{s}=\cfrac{\sqrt{s^2}}{\sqrt{s^2}}=\cfrac{\sqrt[3]{s^3}}{\sqrt[3]{s^3}}\\\\ -------------------------------$

$\bf \cfrac{small}{large}\qquad \stackrel{sides}{\cfrac{s^2}{s^2}}=\stackrel{areas}{\cfrac{6}{150}}\implies \cfrac{\stackrel{\textit{hypotenuse of small}}{5^2}}{\stackrel{\textit{hypotenuse of large}}{s^2}}=\cfrac{6}{150} \\\\\\ \cfrac{5^2\cdot 150}{6}=s^2\implies 625=s^2\implies \sqrt{625}=s\implies \boxed{25=s}\\\\ -------------------------------\\\\ \textit{thus the sides are at a }\cfrac{5}{25}\implies \cfrac{1}{5}\implies 1:5~ratio$

now, notice, the ratio is from small to large, so the larger right-triangle sides are 5 times larger than the small one then.

now, recall, a right-triangle with a hypotenuse of 5, will be a 3,4,5 triangle, you can check using the pythagorean theorem, thus, check the picture below.