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You design a tree house using a plane coordinate plane in which the coordinates are measured in feet. The vertices of the floor are (-2, -3), (-2,4), (5,4), and (5, -3). Find the perimeter (in yards) and the area (in square yards) of the floor.

Respuesta :

Finding the perimeter:

Calculate the lengths of each side:

AB = sqrt((5 - (-2))^2 + (4 - (-3))^2) = 9 feet (using the distance formula)
BC = sqrt((5 - (-2))^2 + (-3 - 4)^2) = 9 feet
CD = sqrt((5 - 5)^2 + (-3 - 4)^2) = 0 feet (diagonally opposite points coincide)
DA = sqrt((-2 - 5)^2 + (-3 - 4)^2) = 9 feet
Add the side lengths to find the perimeter:

Perimeter = AB + BC + CD + DA = 9 ft + 9 ft + 0 ft + 9 ft = 27 feet
Converting perimeter to yards:

Perimeter (yards) = Perimeter (feet) / conversion factor
Conversion factor = 1 yard / 3 feet
Perimeter (yards) = 27 feet * (1 yard / 3 feet) = 9 yards
Finding the area:

Recognize the shape: The four sides are all the same length, and diagonally opposite points coincide, forming a square.

Use the area formula for a square:

Area = side_length^2
Substitute the side length:

Area = 9 feet * 9 feet = 81 square feet
Converting area to square yards:

Area (square yards) = Area (square feet) / conversion factor^2
Conversion factor = 1 yard / 3 feet
Area (square yards) = 81 square feet * (1 yard / 3 feet)^2 = 9 square yards
Therefore, the perimeter of the tree house floor is 9 yards and the area is 9 square yards.

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