A new dog park is being designed by a city planner. The park is enclosed by a fence and shaped like a parallelogram. What is the area and perimeter of the dog park? Round your answ... A new dog park is being designed by a city planner. The park is enclosed by a fence and shaped like a parallelogram. What is the area and perimeter of the dog park? Round your answers to the nearest hundredth, if necessary. Read more

Steps to Solve

1. Calculate the lengths of the sides of the parallelogram

   We use the distance formula to find the lengths of the sides. The distance formula is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. Let's find the length of the side between $(-8, 0)$ and $(-4, -5)$: $d_1 = \sqrt{((-4) - (-8))^2 + ((-5) - 0)^2} = \sqrt{(4)^2 + (-5)^2} = \sqrt{16 + 25} = \sqrt{41}$. Now let's find the length of the side between $(-4, -5)$ and $(9, 0)$: $d_2 = \sqrt{(9 - (-4))^2 + (0 - (-5))^2} = \sqrt{(13)^2 + (5)^2} = \sqrt{169 + 25} = \sqrt{194}$.

2. Calculate the perimeter of the parallelogram

   The perimeter of a parallelogram is $2(a + b)$, where $a$ and $b$ are the lengths of the adjacent sides. In this case, $a = \sqrt{41}$ and $b = \sqrt{194}$. Since 1 unit = 5 meters, we need to multiply each length by 5 to convert it into meters before calculating the perimeter. So, the perimeter is $2(5\sqrt{41} + 5\sqrt{194}) = 10(\sqrt{41} + \sqrt{194}) \approx 10(6.40 + 13.93) = 10(20.33) = 203.3$ meters.

3. Calculate the area of the parallelogram

   We can find the area using the determinant method with the coordinates of the vertices. List the coordinates in a cycle: $(-8, 0)$, $(-4, -5)$, $(9, 0)$, $(5, 5)$, $(-8, 0)$. The area is given by $A = \frac{1}{2} |(x_1y_2 + x_2y_3 + x_3y_4 + x_4y_5) - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_5)|$. $A = \frac{1}{2} |((-8)(-5) + (-4)(0) + (9)(5) + (5)(0)) - ((0)(-4) + (-5)(9) + (0)(5) + (5)(-8))|$ $A = \frac{1}{2} |(40 + 0 + 45 + 0) - (0 - 45 + 0 - 40)| = \frac{1}{2} |85 - (-85)| = \frac{1}{2} |170| = 85$ square units. Since 1 unit = 5 meters, 1 square unit = $5^2 = 25$ square meters. So, the area in square meters is $85 \times 25 = 2125$ square meters.