Question

find the perimeter and total area of the polygon shape shown below.
(image of a polygon composed of a rectangle and two triangles, with side lengths: triangles have sides 10, 10, height 8; rectangle has length 12, width 8)
options:

• perimeter = 96 inches, area = 192 square inches
• perimeter = 56 inches, area = 192 square inches
• perimeter = 96 inches, area = 144 square inches
• perimeter = 56 inches, area = 288 square inches

Explanation:

Step1: Calculate Perimeter

The polygon's perimeter is the sum of all outer sides. The sides are four 10s and two 8s? Wait, no, looking at the shape: top triangle has two sides of 10, middle rectangle has two sides of 8, bottom triangle has two sides of 10. Wait, no, the perimeter: let's list the outer sides. The top triangle: two sides of 10. The middle rectangle: the left and right sides are 8? Wait, no, the middle rectangle is 12 (length) and 8 (height). But the perimeter: the outer edges are the two top triangle sides (10 each), two middle rectangle vertical sides? No, wait, no. Wait, the shape is a top triangle, middle rectangle, bottom triangle. The perimeter is the sum of the outer sides: top triangle's two equal sides (10 each), middle rectangle's left and right? No, wait, no. Wait, the middle rectangle is between the two triangles. So the outer sides: top triangle's two sides (10, 10), middle rectangle's left and right? No, no, the middle rectangle has length 12 and height 8. But the perimeter: the bottom triangle's two sides (10, 10), and the middle rectangle's top and bottom? No, no, the top and bottom of the rectangle are attached to the triangles, so they are not part of the perimeter. Wait, let's visualize: the shape is a composite of a top isoceles triangle (base 12, equal sides 10), a middle rectangle (length 12, height 8), and a bottom isoceles triangle (base 12, equal sides 10). So the perimeter is the sum of the equal sides of the top triangle, the vertical sides of the rectangle? No, wait, no. Wait, the left side: from top triangle's left vertex down the rectangle's left side? No, the rectangle's left side is 8, but the top triangle's left side is 10, and the bottom triangle's left side is 10. Wait, no, the perimeter is the outer edges: top triangle's two sides (10,10), bottom triangle's two sides (10,10), and the two vertical sides of the rectangle? Wait, no, the rectangle's vertical sides: the left and right sides of the rectangle are 8 each? Wait, no, the rectangle has length 12 (horizontal) and height 8 (vertical). So the left and right sides of the rectangle are 8, but are they part of the perimeter? Wait, no, the top triangle is on top of the rectangle (base 12), so the top triangle's base (12) is attached to the rectangle's top (12). Similarly, the bottom triangle's base (12) is attached to the rectangle's bottom (12). So the perimeter is the sum of the top triangle's two equal sides (10,10), the bottom triangle's two equal sides (10,10), and the two vertical sides of the rectangle? Wait, no, the rectangle's left and right sides: the left side is from the top triangle's left vertex down to the bottom triangle's left vertex? Wait, no, the left side of the rectangle is 8, but the top triangle's left side is 10, and the bottom triangle's left side is 10. Wait, I think I made a mistake. Let's count the perimeter: the outer edges are: top triangle: two sides (10,10), bottom triangle: two sides (10,10), and the two vertical sides of the rectangle? No, the rectangle's vertical sides are 8, but the top triangle's left side is 10 (from top vertex to rectangle's left top corner), then the rectangle's left side is 8 (from left top corner to left bottom corner), then the bottom triangle's left side is 10 (from left bottom corner to bottom vertex). Wait, no, that's not right. Wait, the rectangle has length 12 (horizontal) and height 8 (vertical). So the top triangle has base 12 (same as rectangle's top), and equal sides 10. The bottom triangle has base 12 (same as rectangle's bottom), and equal sides 10. So the perimeter is the sum of the top triangle's two equal sides (10,10), the bottom triangle's two equal sides (10,10), and the two vertical sides of the rectangle? Wait, no, the vertical sides of the rectangle: the left and right sides of the rectangle are 8 each, but are they part of the perimeter? Wait, no, the left side of the rectangle is between the top triangle's left vertex and the bottom triangle's left vertex? No, the top triangle's left vertex is connected to the rectangle's top left corner (so the side from top vertex to top left corner is 10), then the rectangle's left side is from top left corner to bottom left corner (length 8), then the bottom triangle's left side is from bottom left corner to bottom vertex (length 10). Similarly on the right. So the perimeter is: top triangle's two sides (10,10), rectangle's left and right sides? No, wait, no. Wait, the perimeter is the total length around the shape. So:

Top triangle: two sides (10,10)

Bottom triangle: two sides (10,10)

Middle rectangle: the left and right sides? Wait, no, the middle rectangle's top and bottom are attached to the triangles, so they are internal. The vertical sides of the rectangle: left and right, each 8? Wait, no, the rectangle has length 12 (horizontal) and height 8 (vertical). So the vertical sides (left and right) are 8 each. But are they part of the perimeter? Wait, yes, because the top triangle is on top (so its base is attached to the rectangle's top, so not part of perimeter), the bottom triangle's base is attached to the rectangle's bottom (not part of perimeter). So the perimeter is:

Top triangle: 10 + 10

Middle rectangle: 8 + 8 (left and right sides)

Bottom triangle: 10 + 10

Wait, that would be 10+10 + 8+8 +10+10 = 10*4 + 8*2 = 40 + 16 = 56. Ah, that makes sense. So perimeter is 56 inches.

Now area:

Area of top triangle: (base * height)/2 = (12 * 8)/2 = 48

Area of middle rectangle: length * height = 12 * 8 = 96

Area of bottom triangle: (base * height)/2 = (12 * 8)/2 = 48

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

So perimeter is 56, area 192.

Step2: Verify Options

Looking at the options:

Option 2: Perimeter = 56 inches, Area = 192 square inches. That matches.

Answer:

B. Perimeter = 56 inches, Area = 192 square inches (assuming the options are labeled as A, B, C, D with the second option being this)