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

Explanation:

Step1: Calculate rectangle's area

The length of the rectangle is $3x + x=4x$ and the width is assumed to be $2x$ (since the vertical segments are divided equally). The area of a rectangle $A_{rectangle}$ is length times width, so $A_{rectangle}=(3x + x)\times2x=4x\times2x = 8x^{2}$.

Step2: Calculate kite's area

The kite can be thought of as composed of two pairs of congruent right - triangles. The area of the kite $A_{kite}$: The two non - overlapping right - triangles that make up the kite have bases and heights that can be used to calculate its area. The area of the kite is half of the area of the rectangle. $A_{kite}=\frac{1}{2}\times A_{rectangle}=\frac{1}{2}\times8x^{2}=4x^{2}$.

Step3: Calculate shaded area

The area of the shaded region $A_{shaded}$ is the area of the rectangle minus the area of the kite. So $A_{shaded}=A_{rectangle}-A_{kite}=8x^{2}-4x^{2}=4x^{2}$.

Answer:

$4x^{2}$