Question

the figure shows a kite inside a rectangle. which expression represents the area of the shaded region? 2x² 4x² 6x² 8x²

Explanation:

Step1: Calculate the area of the rectangle

The length of the rectangle is \(3x + x=4x\), and assume the width is \(2x\) (from the figure's proportion). The area of a rectangle \(A_{rectangle}=\text{length}\times\text{width}=(4x)\times(2x) = 8x^{2}\).

Step2: Calculate the area of the un - shaded region

The un - shaded region can be divided into two parts.

• Part 1: The area of the upper and lower triangles. The base of each triangle is \(3x\) and the height is \(x\). The combined area of the upper and lower triangles \(A_{1}=2\times(\frac{1}{2}\times3x\times x)=3x^{2}\).
• Part 2: The area of the left and right triangles. The base of each triangle is \(x\) and the height is \(2x\). The combined area of the left and right triangles \(A_{2}=2\times(\frac{1}{2}\times x\times2x)=2x^{2}\).

The total area of the un - shaded region \(A_{un - shaded}=A_{1}+A_{2}=3x^{2}+2x^{2}=4x^{2}\).

Step3: Calculate the area of the shaded region

The area of the shaded region \(A_{shaded}=A_{rectangle}-A_{un - shaded}\). Substitute \(A_{rectangle}=8x^{2}\) and \(A_{un - shaded}=4x^{2}\) into the formula, we get \(A_{shaded}=8x^{2}-4x^{2}=4x^{2}\).

Answer:

\(4x^{2}\) (the second option)