Answer:
Part 1) The shape is a trapezoid
Part 2) The perimeter is
or approximately ![135.4\ units](https://tex.z-dn.net/?f=135.4%5C%20units)
Part 3) The area is ![937.5\ units^2](https://tex.z-dn.net/?f=937.5%5C%20units%5E2)
Step-by-step explanation:
step 1
Plot the figure to better understand the problem
we have
A(-28,2),B(-21,-22),C(27,-8),D(-4,9)
using a graphing tool
The shape is a trapezoid
see the attached figure
step 2
Find the perimeter
we know that
The perimeter of the trapezoid is equal to
![P=AB+BC+CD+AD](https://tex.z-dn.net/?f=P%3DAB%2BBC%2BCD%2BAD)
the formula to calculate the distance between two points is equal to
![d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%28y2-y1%29%5E%7B2%7D%2B%28x2-x1%29%5E%7B2%7D%7D)
Find the distance AB
we have
A(-28,2),B(-21,-22)
substitute in the formula
![d=\sqrt{(-22-2)^{2}+(-21+28)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%28-22-2%29%5E%7B2%7D%2B%28-21%2B28%29%5E%7B2%7D%7D)
![d=\sqrt{(-24)^{2}+(7)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%28-24%29%5E%7B2%7D%2B%287%29%5E%7B2%7D%7D)
![d=\sqrt{625}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B625%7D)
![d_A_B=25\ units](https://tex.z-dn.net/?f=d_A_B%3D25%5C%20units)
Find the distance BC
we have
B(-21,-22),C(27,-8)
substitute in the formula
![d=\sqrt{(-8+22)^{2}+(27+21)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%28-8%2B22%29%5E%7B2%7D%2B%2827%2B21%29%5E%7B2%7D%7D)
![d=\sqrt{(14)^{2}+(48)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%2814%29%5E%7B2%7D%2B%2848%29%5E%7B2%7D%7D)
![d=\sqrt{2,500}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B2%2C500%7D)
![d_B_C=50\ units](https://tex.z-dn.net/?f=d_B_C%3D50%5C%20units)
Find the distance CD
we have
C(27,-8),D(-4,9)
substitute in the formula
![d=\sqrt{(9+8)^{2}+(-4-27)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%289%2B8%29%5E%7B2%7D%2B%28-4-27%29%5E%7B2%7D%7D)
![d=\sqrt{(17)^{2}+(-31)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%2817%29%5E%7B2%7D%2B%28-31%29%5E%7B2%7D%7D)
![d=\sqrt{1,250}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B1%2C250%7D)
![d_C_D=25\sqrt{2}\ units](https://tex.z-dn.net/?f=d_C_D%3D25%5Csqrt%7B2%7D%5C%20units)
Find the distance AD
we have
A(-28,2),D(-4,9)
substitute in the formula
![d=\sqrt{(9-2)^{2}+(-4+28)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%289-2%29%5E%7B2%7D%2B%28-4%2B28%29%5E%7B2%7D%7D)
![d=\sqrt{(7)^{2}+(24)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%287%29%5E%7B2%7D%2B%2824%29%5E%7B2%7D%7D)
![d=\sqrt{625}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B625%7D)
![d_A_D=25\ units](https://tex.z-dn.net/?f=d_A_D%3D25%5C%20units)
Find the perimeter
![P=25+50+25\sqrt{2}+25](https://tex.z-dn.net/?f=P%3D25%2B50%2B25%5Csqrt%7B2%7D%2B25)
![P=(100+25\sqrt{2})\ units](https://tex.z-dn.net/?f=P%3D%28100%2B25%5Csqrt%7B2%7D%29%5C%20units)
simplify
----> exact value
![P=135.4\ units](https://tex.z-dn.net/?f=P%3D135.4%5C%20units)
therefore
The perimeter is
or approximately ![135.4\ units](https://tex.z-dn.net/?f=135.4%5C%20units)
step 3
Find the area
The area of trapezoid is equal to
![A=\frac{1}{2}[BC+AD]AB](https://tex.z-dn.net/?f=A%3D%5Cfrac%7B1%7D%7B2%7D%5BBC%2BAD%5DAB)
substitute the given values
![A=\frac{1}{2}[50+25]25=937.5\ units^2](https://tex.z-dn.net/?f=A%3D%5Cfrac%7B1%7D%7B2%7D%5B50%2B25%5D25%3D937.5%5C%20units%5E2)