Question

find the perimeter of a quadrilateral with vertices at c (-2, 1), d (2, 4), e (5, 0), and f (1, -3).

Explanation:

Step1: Use the distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(CD\): \(x_1=-2,y_1 = 1,x_2=2,y_2 = 4\).
\(d_{CD}=\sqrt{(2-(-2))^2+(4 - 1)^2}=\sqrt{(4)^2+(3)^2}=\sqrt{16 + 9}=\sqrt{25}=5\).

Step2: Calculate \(DE\)

For \(DE\): \(x_1=2,y_1 = 4,x_2=5,y_2 = 0\).
\(d_{DE}=\sqrt{(5 - 2)^2+(0 - 4)^2}=\sqrt{(3)^2+(-4)^2}=\sqrt{9+16}=\sqrt{25}=5\).

Step3: Calculate \(EF\)

For \(EF\): \(x_1=5,y_1 = 0,x_2=1,y_2=-3\).
\(d_{EF}=\sqrt{(1 - 5)^2+(-3 - 0)^2}=\sqrt{(-4)^2+(-3)^2}=\sqrt{16 + 9}=\sqrt{25}=5\).

Step4: Calculate \(FC\)

For \(FC\): \(x_1=1,y_1=-3,x_2=-2,y_2 = 1\).
\(d_{FC}=\sqrt{(-2 - 1)^2+(1-(-3))^2}=\sqrt{(-3)^2+(4)^2}=\sqrt{9 + 16}=\sqrt{25}=5\).

Step5: Calculate the perimeter

The perimeter \(P\) of the quadrilateral \(CDEF\) is \(P=d_{CD}+d_{DE}+d_{EF}+d_{FC}\).
Since \(d_{CD}=d_{DE}=d_{EF}=d_{FC} = 5\), then \(P=5 + 5+5+5=20\).

Answer:

20 units