Question

find the perimeter and area of the polygon shown below.

\\(p = 76\text{ feet}, a = 330\text{ square feet}\\)
\\(p = 58\text{ feet}, a = 338\text{ square feet}\\)
\\(p = 76\text{ feet}, a = 338\text{ square feet}\\)
\\(p = 91\text{ feet}, a = 390\text{ square feet}\\)

Explanation:

Identify the components of the polygon

The given polygon is a right trapezoid. We can decompose it into a rectangle on the left and a right triangle on the right by using the dashed vertical height line.

• The height of the trapezoid is \(h = 15\text{ ft}\).
• The bottom base of the rectangle is \(18\text{ ft}\), which means the top side of the rectangle is also \(18\text{ ft}\).
• The right triangle has a horizontal leg of \(8\text{ ft}\) at the top and a hypotenuse of \(17\text{ ft}\).
• The vertical leg of this right triangle is equal to the height of the trapezoid, which is \(15\text{ ft}\).

Find the missing side lengths

Let us verify the right triangle dimensions using the Pythagorean theorem:

$$ 15^2 + 8^2 = 225 + 64 = 289 = 17^2 $$

This confirms the dimensions are consistent.

• The total top base of the trapezoid is:

$$ b_1 = 18\text{ ft} + 8\text{ ft} = 26\text{ ft} $$

• The bottom base of the trapezoid is:

$$ b_2 = 18\text{ ft} $$

• The left vertical side is \(15\text{ ft}\).
• The right slanted side is \(17\text{ ft}\).

Calculate the perimeter

The perimeter \(P\) is the sum of all outer boundary sides of the trapezoid:

$$ P = \text{left side} + \text{bottom base} + \text{slanted side} + \text{top base} $$

$$ P = 15 + 18 + 17 + 26 = 76\text{ ft} $$

Calculate the area

The area \(A\) can be calculated as the sum of the areas of the rectangle and the right triangle:

• Area of the rectangle:

$$ A_{\text{rect}} = \text{width} \times \text{height} = 18 \times 15 = 270\text{ sq ft} $$

• Area of the right triangle:

$$ A_{\text{tri}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \times 15 = 60\text{ sq ft} $$

• Total Area:

$$ A = A_{\text{rect}} + A_{\text{tri}} = 270 + 60 = 330\text{ sq ft} $$

Alternatively, using the trapezoid area formula:

$$ A = \frac{b_1 + b_2}{2} \times h = \frac{26 + 18}{2} \times 15 = 22 \times 15 = 330\text{ sq ft} $$

Match with the options

• \(P = 76\text{ feet}\)
• \(A = 330\text{ square feet}\)

This matches the first option.

Answer:

• (A) \(P = 76\text{ feet}, A = 330\text{ square feet}\) (Correct answer)
• (B) \(P = 58\text{ feet}, A = 338\text{ square feet}\)
• (C) \(P = 76\text{ feet}, A = 338\text{ square feet}\)
• (D) \(P = 91\text{ feet}, A = 390\text{ square feet}\)