Question

what is the perimeter of kite wxyz? 2\sqrt{53}+2 units 2\sqrt{53}+5 units 2\sqrt{53}+10 units 2\sqrt{53}+14 units

Explanation:

Step1: Calculate the length of \(WZ\)

Since \(W(-3,3)\) and \(Z(-3,-2)\), using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), here \(x_1=-3,y_1 = 3,x_2=-3,y_2=-2\). Then \(d=\sqrt{(-3+3)^2+(-2 - 3)^2}=\sqrt{0+(-5)^2}=5\)

Step2: Calculate the length of \(XY\)

Since \(X(2,3)\) and \(Y(4,-4)\), using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), here \(x_1 = 2,y_1=3,x_2=4,y_2=-4\). Then \(d=\sqrt{(4 - 2)^2+(-4 - 3)^2}=\sqrt{4 + 49}=\sqrt{53}\)

Step3: Calculate the length of \(WX\)

Since \(W(-3,3)\) and \(X(2,3)\), using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), here \(x_1=-3,y_1 = 3,x_2=2,y_2=3\). Then \(d=\sqrt{(2 + 3)^2+(3 - 3)^2}=\sqrt{25+0}=5\)

Step4: Calculate the length of \(YZ\)

Since \(Y(4,-4)\) and \(Z(-3,-2)\), using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), here \(x_1 = 4,y_1=-4,x_2=-3,y_2=-2\). Then \(d=\sqrt{(-3 - 4)^2+(-2 + 4)^2}=\sqrt{49+4}=\sqrt{53}\)

Step5: Calculate the perimeter of the kite

The perimeter \(P\) of a kite is \(P = 2(WZ+XY)\) (because in a kite \(WZ = WX\) and \(XY=YZ\)). Substitute \(WZ = 5\) and \(XY=\sqrt{53}\) into the formula, \(P=2(5+\sqrt{53})=2\sqrt{53}+10\)

Answer:

\(2\sqrt{53}+10\) units