Question

find the perimeter and area of this figure.

round your answer to the nearest hundredth.

Explanation:

Step1: Label the points

Let the points be \(A(1,5)\), \(B(4,8)\), \(C(7,5)\), \(D(4,2)\)

Step2: Calculate the distance between two points

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)
For \(AB\): \(x_1 = 1,y_1 = 5,x_2 = 4,y_2 = 8\)
\(AB=\sqrt{(4 - 1)^2+(8 - 5)^2}=\sqrt{9 + 9}=\sqrt{18}\approx4.24\)
For \(BC\): \(x_1 = 4,y_1 = 8,x_2 = 7,y_2 = 5\)
\(BC=\sqrt{(7 - 4)^2+(5 - 8)^2}=\sqrt{9 + 9}=\sqrt{18}\approx4.24\)
For \(CD\): \(x_1 = 7,y_1 = 5,x_2 = 4,y_2 = 2\)
\(CD=\sqrt{(4 - 7)^2+(2 - 5)^2}=\sqrt{9 + 9}=\sqrt{18}\approx4.24\)
For \(DA\): \(x_1 = 4,y_1 = 2,x_2 = 1,y_2 = 5\)
\(DA=\sqrt{(1 - 4)^2+(5 - 2)^2}=\sqrt{9 + 9}=\sqrt{18}\approx4.24\)

Step3: Calculate the perimeter

\(P=AB + BC+CD + DA\)
\(P = 4\times\sqrt{18}\approx4\times4.24 = 16.97\)

Step4: Calculate the area

Use the formula for the area of a kite \(A=\frac{1}{2}\times d_1\times d_2\)
The diagonals: \(d_1=(8 - 2)=6\), \(d_2=(7 - 1)=6\)
\(A=\frac{1}{2}\times6\times6 = 18\)

Answer:

$P = 16.97$ units, $A = 18$ units²