Question

find the perimeter and total area of the polygon shape shown below. perimeter = 96 inches, area = 144 square inches perimeter = 56 inches, area = 192 square inches perimeter = 56 inches, area = 288 square inches perimeter = 96 inches, area = 192 square inches

Explanation:

Step1: Calculate the Perimeter

The polygon is composed of 4 sides of length 10 and 2 sides of length 12? Wait, no, looking at the figure: the top triangle has two sides of 10, the rectangle has two sides of 8 and one of 12? Wait, no, let's list all the outer sides. The figure has four sides of 10 (top triangle two sides, bottom triangle two sides) and four sides? Wait, no, the perimeter is the sum of all the outer edges. Let's see: top triangle: two sides of 10. Then the rectangle: but the rectangle's top and bottom are connected to the triangles, so the outer sides of the rectangle are the left and right sides? Wait, no, the figure is a combination of a top triangle, a rectangle, and a bottom triangle. The outer edges: top triangle has two sides of 10, bottom triangle has two sides of 10, and the left and right sides? Wait, no, the rectangle has length 12 and height 8? Wait, no, the rectangle's sides: left side 8, right side 8? No, the rectangle is 12 (length) and 8 (height)? Wait, no, the figure: top triangle (base 12, height 8), rectangle (length 12, height 8), bottom triangle (base 12, height 8). The perimeter: the outer sides are the four sides of 10 (top triangle two, bottom triangle two) and the two sides of 8? No, wait, no. Wait, the left side: from top triangle's left vertex down to the bottom triangle's left vertex: that's 8 (rectangle's left side) plus... no, the rectangle's left side is 8, right side is 8? No, the rectangle is 12 (length) and 8 (height). Wait, no, the figure: the top triangle has two equal sides (10 each), base 12. The rectangle has length 12, height 8. The bottom triangle has two equal sides (10 each), base 12. So the perimeter is the sum of the four sides of 10 (top two, bottom two) and the two sides of 8? No, wait, no. Wait, the left side: from top triangle's left vertex, down the rectangle's left side (8) to the bottom triangle's left vertex, then up the bottom triangle's left side (10)? No, no, the perimeter is the outer edges. Let's count: top triangle: two sides of 10. Then the right side: from top triangle's right vertex, down the rectangle's right side (8) to the bottom triangle's right vertex, then up the bottom triangle's right side (10). Then the left side: from top triangle's left vertex, down the rectangle's left side (8) to the bottom triangle's left vertex, then up the bottom triangle's left side (10). Wait, no, that can't be. Wait, maybe I'm wrong. Let's look at the options. The options have perimeter 56 or 96. Let's calculate: the figure has four sides of 10 (top triangle two, bottom triangle two) and four sides? Wait, no, the rectangle's top and bottom are internal (connected to the triangles), so the outer sides are: top triangle: two sides of 10. Rectangle: left and right sides? No, the rectangle's left and right sides are 8? Wait, no, the rectangle is 12 (length) and 8 (height). Wait, the left side of the rectangle is 8, right side is 8. Then the bottom triangle's two sides of 10, top triangle's two sides of 10. Wait, no, that would be 10 + 10 + 8 + 10 + 10 + 8? No, that's 56. Wait, 10*4 + 8*2? 10*4=40, 8*2=16, 40+16=56. So perimeter is 56 inches.

Step2: Calculate the Area

The area is the sum of the area of the top triangle, the rectangle, and the bottom triangle.

Area of a triangle: $\frac{1}{2} \times base \times height$.

Top triangle: base 12, height 8. Area: $\frac{1}{2} \times 12 \times 8 = 48$ square inches.

Rectangle: length 12, height 8. Area: $12 \times 8 = 96$ square inches.

Bottom triangle: base 12, height 8. Area: $\frac{1}{2} \times 12 \times 8 = 48$ square inches.

Total area: $48 + 96 + 48 = 192$ square inches.

Answer:

Perimeter = 56 inches, Area = 192 square inches (corresponding to the option: Perimeter = 56 inches, Area = 192 square inches)