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 two 5 in. by 1.5 in. and two 2 in. by 1.5 in. 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 21 square inches.
The Correct Answer and Explanation is:
To determine the area of the shaded part of the figure, we can analyze it in two different ways—either by decomposing it into rectangles or into trapezoids. Based on the dimensions provided:
- 5 in.
- 2 in.
- 2 in.
- 5 in.
And given that the total area is 21 square inches, here’s how to proceed:
Correct Drop-down Answers:
The figure can be decomposed into two 5 in. by 1.5 in. and two 2 in. by 1.5 in. rectangles,
or
the figure can be decomposed into 4 trapezoids with heights of 1.5 in. and bases of 2 in. and 5 in.
Explanation
To find the area of the shaded figure, we can use two effective decomposition strategies: breaking it into rectangles or into trapezoids.
Method 1: Decomposing into Rectangles
According to the figure’s dimensions, the shape includes:
- Two 5-inch by 1.5-inch rectangles
- Two 2-inch by 1.5-inch rectangles
To compute the total area, find the area of each rectangle and sum them up:
- Area of each 5 in. × 1.5 in. rectangle = 5 × 1.5 = 7.5 square inches
- Area of each 2 in. × 1.5 in. rectangle = 2 × 1.5 = 3 square inches
Now, total area:
- 2 × 7.5 = 15 square inches
- 2 × 3 = 6 square inches
- Total area = 15 + 6 = 21 square inches
Method 2: Decomposing into Trapezoids
Alternatively, the figure can be split into four trapezoids, each with:
- A height of 1.5 inches
- Bases of 2 inches and 5 inches
Use the area formula for a trapezoid:Area=12×(Base1+Base2)×Height\text{Area} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height}Area=21×(Base1+Base2)×Height=12×(2+5)×1.5=12×7×1.5=5.25 square inches= \frac{1}{2} \times (2 + 5) \times 1.5 = \frac{1}{2} \times 7 \times 1.5 = 5.25\ \text{square inches}=21×(2+5)×1.5=21×7×1.5=5.25 square inches
Each trapezoid is 5.25 square inches, and with 4 trapezoids:4×5.25=21 square inches4 \times 5.25 = 21\ \text{square inches}4×5.25=21 square inches
Both methods yield the same total shaded area of 21 square inches, verifying the accuracy of the decomposition.
