The Correct Answer and Explanation is:
Correct Answer
The missing side x is 10 inches.
The perimeter P is 138 inches.
The area A is 832 square inches.
Explanation
To find the required values, we will first determine the length of the missing side x, then use it to calculate the perimeter and area of the entire figure.
Finding the Missing Side (x)
The shape provided is a rectilinear polygon, meaning all its interior angles are 90 or 270 degrees and its sides are straight. In such a figure, the vertical column, or stem, of the cross has a constant width. The diagram indicates that the width of the stem at the top is 10 inches. Because the sides of the stem are parallel, its width must be the same along its entire length. Therefore, the width at the bottom of the stem, labeled as x, must also be 10 inches.
Calculating the Perimeter (P)
The perimeter is the total distance around the outside of a shape. To find it, we sum the lengths of all the sides. Now that we know x = 10 inches, we can add all the known side lengths together:
P = 10 + 11 + 11 + 16 + 21 + x + 21 + 16 + 11 + 11
Substituting 10 for x:
P = 10 + 11 + 11 + 16 + 21 + 10 + 21 + 16 + 11 + 11 = 138 inches.
So, the perimeter of the figure is 138 inches.
Calculating the Area (A)
To calculate the area of this complex shape, we can divide it into simpler rectangular sections. One effective method is to treat it as one long vertical rectangle and two horizontal rectangular “arms.”
- Vertical Rectangle: This central column has a width of 10 inches. Its total height is the sum of the vertical sections on one side: 11 in + 16 in + 21 in = 48 inches.
Area of vertical rectangle = width × height = 10 in × 48 in = 480 square inches. - Horizontal Arms: The two arms extend from the middle section. Each arm has a length of 11 inches and a height of 16 inches.
Area of one arm = 11 in × 16 in = 176 square inches.
Since there are two identical arms, their total area is 2 × 176 = 352 square inches. - Total Area: The total area of the cross is the sum of the areas of these parts.
Total Area A = 480 sq in + 352 sq in = 832 square inches.
