Question

find the perimeter, in feet, of the parallelogram.
options: 82 feet, 44 feet, 88 feet, 68 feet (diagram of parallelogram with right triangle showing 24 ft, 17 ft, 27 ft)

Explanation:

Step1: Identify the sides of the parallelogram

In a parallelogram, opposite sides are equal. From the diagram, we can see that one pair of sides has length \( 27 \) ft and the other pair can be found using the right triangle (since there's a right angle, we can use the Pythagorean theorem? Wait, no, wait. Wait, the right triangle has legs \( 8 \) ft (wait, the vertical side? Wait, the diagram shows a right triangle with one leg \( 8 \) ft? Wait, no, the given sides: wait, the parallelogram has a side of \( 27 \) ft, and the other side: wait, the right triangle has legs \( 8 \) ft and... Wait, no, maybe I misread. Wait, the problem: the parallelogram has a side of \( 17 \) ft? Wait, no, the diagram: let's re-express. Wait, the right triangle has a horizontal leg of \( 8 \) ft? Wait, no, the vertical side? Wait, maybe the other side of the parallelogram is calculated as follows: Wait, no, maybe the sides are \( 27 \) ft and \( 17 + 8 \)? Wait, no, wait. Wait, the right triangle: the horizontal leg is \( 8 \) ft, the vertical leg? Wait, no, the parallelogram: in a parallelogram, the perimeter is \( 2(a + b) \), where \( a \) and \( b \) are the lengths of adjacent sides. From the diagram, one side is \( 27 \) ft, and the other side: the right triangle has a leg of \( 8 \) ft and the other leg? Wait, no, the side of the parallelogram: wait, the vertical side of the right triangle is \( 15 \)? Wait, no, maybe I made a mistake. Wait, the options are 82, 44, 88, 68. Let's check:

Wait, maybe the sides are \( 27 \) ft and \( 17 + 8 \)? No, wait, the right triangle: the horizontal leg is \( 8 \) ft, and the other leg (the side of the parallelogram) is \( 17 \)? Wait, no, maybe the adjacent sides are \( 27 \) ft and \( 17 + 8 \)? Wait, no, let's calculate:

Wait, the perimeter of a parallelogram is \( 2 \times ( \text{length of one side} + \text{length of adjacent side} ) \).

From the diagram, one side is \( 27 \) ft. The other side: the right triangle has legs \( 8 \) ft and \( 15 \) ft? Wait, no, \( 8^2 + 15^2 = 17^2 \)? Wait, \( 8^2 = 64 \), \( 15^2 = 225 \), \( 64 + 225 = 289 = 17^2 \). Oh! So the right triangle has legs \( 8 \) ft and \( 15 \) ft? Wait, no, the hypotenuse would be \( 17 \) ft. Wait, maybe the side of the parallelogram is \( 17 + 8 \)? No, wait, the horizontal leg is \( 8 \) ft, and the other side of the parallelogram is \( 17 \) ft? Wait, no, let's re-express.

Wait, the diagram: the parallelogram has a side of \( 27 \) ft, and the other side: the right triangle is formed by dropping a height, so the base of the right triangle is \( 8 \) ft, and the hypotenuse of the right triangle is the side of the parallelogram? Wait, no, the side of the parallelogram adjacent to \( 27 \) ft: let's see, the vertical side of the right triangle is \( 15 \) ft (since \( 17^2 - 8^2 = 289 - 64 = 225 = 15^2 \)), but that might not be relevant. Wait, maybe the adjacent sides are \( 27 \) ft and \( 17 \) ft? No, that doesn't make sense. Wait, the options: let's check the perimeter.

Wait, maybe the sides are \( 27 \) ft and \( 17 \) ft? No, \( 2(27 + 17) = 2(44) = 88 \). Oh! Wait, \( 27 + 17 = 44 \), times 2 is 88. So the perimeter is \( 2 \times (27 + 17) = 88 \) feet? Wait, but where does the 8 ft come into play? Wait, maybe the 8 ft is a distractor, or maybe I misread the diagram. Wait, the diagram shows a right triangle with one leg 8 ft, and the other side of the parallelogram is 17 ft? Wait, no, maybe the adjacent sides are 27 ft and (17 + 8)? No, 17 + 8 is 25, 2(27 + 25) = 104, which is not an option. Wait, the options are 82, 44, 88, 68. So 88 is an option. Let's check: 27 + 17 = 44, times 2 is 88. So maybe the 8 ft is a mistake, or maybe the diagram has 17 ft as one side, 27 ft as the other, so perimeter is 2*(27 + 17) = 88.

Step2: Calculate the perimeter

The formula for the perimeter \( P \) of a parallelogram is \( P = 2(a + b) \), where \( a \) and \( b \) are the lengths of adjacent sides.

From the diagram, let \( a = 27 \) ft and \( b = 17 \) ft (assuming the 8 ft is a part of the right triangle but the adjacent side is 17 ft). Then:

\( P = 2(27 + 17) \)

\( P = 2(44) \)

\( P = 88 \) feet.

Answer:

88 feet