Question

A kite is inscribed in a rectangle. Everything outside of the kite is shaded. The points of the kite split the lengths of the rectangle into a length of 3x and x. The points of the kite split the height of the rectangle into lengths of x and x. The total length of the rectangle is 4 x and the height is 2 x.
The figure shows a kite inside a rectangle. Which expression represents the area of the shaded region?

2x2
4x2
6x2

Answer 1

Rectangle area = (4x)(2x) = 8x^2.

The shaded region is the four corner right triangles. Their areas:
- top-left: (1/2)(3x)(x) = 3/2 x^2
- top-right: (1/2)(1x)(x) = 1/2 x^2
- bottom-right: (1/2)(3x)(x) = 3/2 x^2
- bottom-left: (1/2)(1x)(x) = 1/2 x^2

Sum = 3/2 + 1/2 + 3/2 + 1/2 = 4x^2.

Thus the shaded area is 4x^2.