Question

the area of the shaded part of the figure below can be determined in different ways. use the drop-down menus to explain how to find the area by decomposing the figure into rectangles or trapezoids. 5 in. 2 in. 2 in. 5 in. click the arrows to choose an answer from each menu. the figure can be decomposed into choose... rectangles, or the figure can be decomposed into 4 trapezoids with heights of choose... in. and bases of choose... in. and 5 in. using either method, the total area of the shaded figure is choose... square inches.

Explanation:

Step1: Calculate the area of the large square

The area of a square is \(A = s^2\), where \(s = 5\) in. So \(A_{large}=5^2=25\) square inches.

Step2: Calculate the area of the small square

The side - length of the small square \(s = 2\) in. So \(A_{small}=2^2 = 4\) square inches.

Step3: Calculate the area of the shaded region

The area of the shaded region \(A=A_{large}-A_{small}\). Substitute the values: \(A = 25-4=21\) square inches.

If we consider the trapezoid method:
The formula for the area of a trapezoid is \(A=\frac{(b_1 + b_2)h}{2}\). When decomposing into 4 trapezoids, the height \(h=\frac{5 - 2}{2}=1.5\) in, and the two bases \(b_1 = 2\) in and \(b_2 = 5\) in.
The area of one trapezoid \(A_1=\frac{(2 + 5)\times1.5}{2}=\frac{7\times1.5}{2}=5.25\) square inches.
The area of 4 trapezoids \(A = 4\times A_1=4\times5.25 = 21\) square inches.

Answer:

The figure can be decomposed into \(5\) rectangles (by considering the large square as one and subtracting the small square, or more precisely, if we think of the border as composed of rectangles around the small square. For example, 4 rectangles of size \(1.5\times2\) and 1 square of \(2\times2\) within the non - shaded part, but a simpler way is using the area of the large square minus the small square). Or the figure can be decomposed into 4 trapezoids with heights of \(1.5\) in. and bases of \(2\) in. and \(5\) in. Using either method, the total area of the shaded figure is \(21\) square inches.