Question

Let \( ABC \) be a triangle right-angled at \( B \) with \( AB = BC = 18 \). The area of the largest rectangle that can be inscribed in this triangle and has \( B \) as one of the vertices is  _____________.

Hint:

In right-angled triangles, the largest inscribed rectangle has one vertex at the right angle, and its area is half of the area of the triangle.

Answer 1

Step 1: Use the geometry of the triangle. In a right-angled triangle, the largest rectangle that can be inscribed with one vertex at the right angle is half the area of the triangle. Given that the triangle is right-angled at \( B \), with \( AB = BC = 18 \), the area of the triangle is: \[ \text{Area of triangle} = \frac{1}{2} \times AB \times BC = \frac{1}{2} \times 18 \times 18 = 162 \] Step 2: Find the area of the largest inscribed rectangle. The area of the largest rectangle inscribed in the right-angled triangle is half the area of the triangle: \[ \text{Area of rectangle} = \frac{1}{2} \times 162 = 81 \] Thus, the area of the largest rectangle is: \[ \boxed{81} \]