Question

Suppose that a right triangle has legs of lengths 9 cm and 3 cm. A rectangle is inscribed in this right triangle so that two sides of the rectangle lie along the legs. Find the largest possible area of such a rectangle.

Answer 1

The largest possible area of a rectangle inscribed in the right triangle is 6 cm², achieved with rectangle side lengths of 6 cm and 1 cm.

Step-by-step explanation:

To find the largest possible area of a rectangle inscribed in a right triangle with legs of lengths 9 cm and 3 cm, we can use calculus to maximize the area function or geometric reasoning. Let the rectangle sides be x and y, aligned with the triangle legs. As the rectangle is inscribed in the right triangle, a smaller similar right triangle with legs (9 - x) and (3 - y) forms adjacent to the rectangle. By the similarity of triangles, the ratio of corresponding sides is constant, so \((9 - x)/9 = (3 - y)/3\). Simplifying this, we get \(y = 3 - x/3\).

The area of the rectangle is A = xy, substituting for y, we get A(x) = x(3 - x/3). To find the maximum area, we can take the derivative of A with respect to x and set it to zero to find the critical points. Solving this, we find that the largest area occurs when x = 6 cm and y = 1 cm, making the maximum area 6 cm².