To find the perimeter and area of trapezoid WXYZ given the vertices W(-1, 6), X(1, 6), Y(7, -2), and Z(-7, -2), we can follow these steps:
Step 1: Calculate the side lengths
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WX: We can calculate the length of WX using the formula for the distance between two points: \[ WX = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(1 - (-1))^2 + (6 - 6)^2} = \sqrt{(2)^2 + (0)^2} = \sqrt{4} = 2 \]
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XY: \[ XY = \sqrt{(7 - 1)^2 + (-2 - 6)^2} = \sqrt{(6)^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \]
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YZ: \[ YZ = \sqrt{(-7 - 7)^2 + (-2 - (-2))^2} = \sqrt{(-14)^2 + (0)^2} = \sqrt{196} = 14 \]
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ZW: \[ ZW = \sqrt{(-1 - (-7))^2 + (6 - (-2))^2} = \sqrt{(6)^2 + (8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \]
Step 2: Calculate the perimeter
The perimeter \(P\) of the trapezoid is the sum of the lengths of all its sides: \[ P = WX + XY + YZ + ZW = 2 + 10 + 14 + 10 = 36 \text{ units} \]
Since 1 unit = 12 feet: \[ P = 36 \text{ units} \times 12 \text{ feet/unit} = 432 \text{ feet} \]
Step 3: Calculate the area
To find the area of trapezoid WXYZ, we can use the formula for the area of a trapezoid: \[ \text{Area} = \frac{1}{2} \times (b_1 + b_2) \times h \] where \(b_1\) and \(b_2\) are the lengths of the two parallel sides, and \(h\) is the height.
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The lengths of the parallel sides are:
- \(b_1 = WX = 2\)
- \(b_2 = YZ = 14\)
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The height \(h\) can be found by calculating the vertical distance between the y-coordinates of the two horizontal sides (\(y = 6\) and \(y = -2\)): \[ h = 6 - (-2) = 8 \]
Now we can substitute these values into the area formula: \[ \text{Area} = \frac{1}{2} \times (2 + 14) \times 8 = \frac{1}{2} \times 16 \times 8 = 8 \times 8 = 64 \text{ square units} \]
Since one unit square = \(12 \text{ feet} \times 12 \text{ feet}\): \[ \text{Area} = 64 \text{ square units} \times (12 \text{ feet/unit})^2 = 64 \times 144 = 9216 \text{ square feet} \]
Final Answer
- Perimeter: 432 feet
- Area: 9216 square feet
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