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.
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.
Solution
Step 1: Area of the Right Triangle
The total area of right triangle \(ABC\) is:
\[ \text{Area of } \triangle ABC = \frac{1}{2} \times 40 \times 30 = 600 \]Step 2: Area Relationships & Equation Setup
Let \(x\) be the side length of the square.
Area of triangle \(BEC\):
\[ \frac{1}{2} \times 40 \times x = 20x \]Area of triangle \(BEA\):
\[ \frac{1}{2} \times 30 \times x = 15x \]Equating the sum of these constituent areas to the total area of triangle \(ABC\):
\[ 20x + 15x = 600 \]Step 3: Solving for \(x\)
Therefore, the side length of the inscribed square is:
\[ \boxed{x = \frac{120}{7} \approx 17.14} \]More Geometry Resources
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