Question

finding the perimeter of a kite in the coordinate plane
a sandbox is shaped like a kite. a planner would like to replace the wooden border around the sandbox. how many feet of wood does he need? round up to the nearest whole number.
feet

Explanation:

Step1: Find the lengths of the sides using the distance formula

The distance formula is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
Let the vertices of the kite (sand - box) be \(A(0,6)\), \(B(8,0)\), \(C(14,6)\) and \(D(8,12)\)
For side \(AB\): \(x_1 = 0,y_1=6,x_2 = 8,y_2 = 0\)
\(d_{AB}=\sqrt{(8 - 0)^2+(0 - 6)^2}=\sqrt{64 + 36}=\sqrt{100}=10\)
For side \(BC\): \(x_1 = 8,y_1 = 0,x_2=14,y_2 = 6\)
\(d_{BC}=\sqrt{(14 - 8)^2+(6 - 0)^2}=\sqrt{36+36}=\sqrt{72}=6\sqrt{2}\approx8.49\)
Since a kite has two pairs of adjacent sides equal, \(AB = AD = 10\) and \(BC=CD = 6\sqrt{2}\approx8.49\)

Step2: Calculate the perimeter

The perimeter \(P\) of a kite is \(P=2(AB + BC)\)
\(P = 2(10+6\sqrt{2})\)
\(P=20 + 12\sqrt{2}\)
\(P\approx20+12\times1.414=20 + 16.97=36.97\)

Answer:

\(37\)