Question

bella lives in a cul de sac in austin. Trapezoid WXYZ represents her back yard. Given 1 unit =12 feet find the perimeter of the backyard and the area
W(-1,6) X (1,6) Y(7,-2) Z(-7,-2)

Answer 1

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

1. **WX**: We can calculate the length of WX using the formula for the distance between two points:
\[
WX = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(1 - (-1))^2 + (6 - 6)^2} = \sqrt{(2)^2 + (0)^2} = \sqrt{4} = 2
\]

2. **XY**:
\[
XY = \sqrt{(7 - 1)^2 + (-2 - 6)^2} = \sqrt{(6)^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10
\]

3. **YZ**:
\[
YZ = \sqrt{(-7 - 7)^2 + (-2 - (-2))^2} = \sqrt{(-14)^2 + (0)^2} = \sqrt{196} = 14
\]

4. **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.

1. The lengths of the parallel sides are:
- \(b_1 = WX = 2\)
- \(b_2 = YZ = 14\)

2. 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**