Question

1. find the perimeter and area of the given trapezoid.

Explanation:

Step1: Identify the lengths of the sides

Let the vertices of the trapezoid be \(A(-5,6)\), \(B(-3,2)\), \(C(1,2)\), \(D(3,6)\).
Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(AB\): \(x_1=-5,y_1 = 6,x_2=-3,y_2 = 2\), \(AB=\sqrt{(-3 + 5)^2+(2 - 6)^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}\).
For \(BC\): \(x_1=-3,y_1 = 2,x_2=1,y_2 = 2\), \(BC=\sqrt{(1+ 3)^2+(2 - 2)^2}=4\).
For \(CD\): \(x_1=1,y_1 = 2,x_2=3,y_2 = 6\), \(CD=\sqrt{(3 - 1)^2+(6 - 2)^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}\).
For \(DA\): \(x_1=3,y_1 = 6,x_2=-5,y_2 = 6\), \(DA=\sqrt{(-5 - 3)^2+(6 - 6)^2}=8\).

Step2: Calculate the perimeter \(P\)

The perimeter of a trapezoid \(P=AB + BC+CD + DA\).
\(P=2\sqrt{5}+4 + 2\sqrt{5}+8=12 + 4\sqrt{5}\).

Step3: Calculate the area \(A\)

The formula for the area of a trapezoid is \(A=\frac{(b_1 + b_2)h}{2}\), where \(b_1\) and \(b_2\) are the lengths of the parallel - sides and \(h\) is the height.
The parallel sides are \(b_1 = 4\) and \(b_2=8\), and the height \(h = 4\).
\(A=\frac{(4 + 8)\times4}{2}=24\).

Answer:

Perimeter: \(12 + 4\sqrt{5}\), Area: \(24\)