Problem
In the figure below, BDEF is a square inscribed in right triangle ABC. The legs of the right triangle measure 40 and 30 units. Find the length \(x\) of the side of the square.
In this geometry problem, we determine the side length of a square inscribed inside a right triangle. Using area relationships, we derive an equation and solve it step-by-step.
In the figure below, BDEF is a square inscribed in right triangle ABC. The legs of the right triangle measure 40 and 30 units. Find the length \(x\) of the side of the square.
The total area of right triangle \(ABC\) is:
\[ \text{Area of } \triangle ABC = \frac{1}{2} \times 40 \times 30 = 600 \]Let \(x\) be the side length of the square. The two triangles outside the square, \( \triangle BEC \) and \( \triangle BEA \), together have the same total area as the original triangle.
Area of triangle \(BEC\):
\[ \frac{1}{2} \times 40 \times x = 20x \]Area of triangle \(BEA\):
\[ \frac{1}{2} \times 30 \times x = 15x \]Since their combined area equals 600, we write:
\[ 20x + 15x = 600 \] \[ 35x = 600 \] \[ x = \frac{600}{35} = \frac{120}{7} \] \[ x \approx 17.14 \]Therefore, the side length of the inscribed square is:
\[ \boxed{x = \frac{120}{7} \approx 17.14} \]Explore additional tutorials, worked problems, and interactive applets here: