3. The shorter leg of a 30^circ -60^circ -90^circ triangle is 8.2 inches long. What is the perimeter of the right triangle? Round your answer to the nearest tenth. A. 36.2 in B. 38.8 in C. 426 in D. 44.3 in

To solve this problem, we need to find the lengths of the other two sides of the triangle and then calculate the perimeter.<br /><br />Given information:<br />- The shorter leg of the triangle is 8.2 inches long.<br />- The triangle is a $30^{\circ }-60^{\circ }-90^{\circ }$ triangle.<br /><br />In a $30^{\circ }-60^{\circ }-90^{\circ }$ triangle, the ratio of the sides is:<br />- Shorter leg (opposite $30^{\circ }$) : Longer leg (opposite $60^{\circ }$) : Hypotenuse = 1 : $\sqrt{3}$ : 2<br /><br />Step 1: Find the length of the longer leg.<br />Longer leg = Shorter leg × $\sqrt{3}$<br />Longer leg = 8.2 × $\sqrt{3}$<br />Longer leg ≈ 14.2 inches<br /><br />Step 2: Find the length of the hypotenuse.<br />Hypotenuse = 2 × Shorter leg<br />Hypotenuse = 2 × 8.2<br />Hypotenuse = 16.4 inches<br /><br />Step 3: Calculate the perimeter of the triangle.<br />Perimeter = Shorter leg + Longer leg + Hypotenuse<br />Perimeter = 8.2 + 14.2 + 16.4<br />Perimeter ≈ 38.8 inches<br /><br />Therefore, the correct answer is B. 38.8 in.