Question

quadrilateral klmn has vertices with the coordinates, k(5, -2), l(-3, 1), m(2, 4), and n(7, 4). find the lengths of sides kl, lm, mn and nk, and finally the perimeter of quadrilateral klmn.
drag and drop the answers to the boxes provided to complete the solution.
the length of kl is units.
the length of lm is units.
the length of mn is units.
the length of nk is units
the perimeter of quadrilateral klmn is units

Explanation:

Step1: Use the distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

Step2: Calculate the length of \(KL\)

For \(K(5,-2)\) and \(L(-3,1)\)
\(x_1 = 5,y_1=-2,x_2=-3,y_2 = 1\)
\(KL=\sqrt{(-3 - 5)^2+(1+ 2)^2}=\sqrt{(-8)^2+3^2}=\sqrt{64 + 9}=\sqrt{73}\approx8.54\)

Step3: Calculate the length of \(LM\)

For \(L(-3,1)\) and \(M(2,4)\)
\(x_1=-3,y_1 = 1,x_2=2,y_2=4\)
\(LM=\sqrt{(2 + 3)^2+(4 - 1)^2}=\sqrt{5^2+3^2}=\sqrt{25+9}=\sqrt{34}\approx5.83\)

Step4: Calculate the length of \(MN\)

For \(M(2,4)\) and \(N(7,4)\)
\(x_1=2,y_1 = 4,x_2=7,y_2=4\)
\(MN=\sqrt{(7 - 2)^2+(4 - 4)^2}=\sqrt{5^2+0^2}=5\)

Step5: Calculate the length of \(NK\)

For \(N(7,4)\) and \(K(5,-2)\)
\(x_1=7,y_1 = 4,x_2=5,y_2=-2\)
\(NK=\sqrt{(5 - 7)^2+(-2 - 4)^2}=\sqrt{(-2)^2+(-6)^2}=\sqrt{4 + 36}=\sqrt{40}=2\sqrt{10}\approx6.32\)

Step6: Calculate the perimeter

Perimeter \(P=KL + LM+MN+NK\)
\(P=\sqrt{73}+\sqrt{34}+5 + 2\sqrt{10}\approx8.54+5.83+5+6.32=25.69\)

Answer:

The length of \(KL\) is \(\sqrt{73}\approx8.54\) units.
The length of \(LM\) is \(\sqrt{34}\approx5.83\) units.
The length of \(MN\) is \(5\) units.
The length of \(NK\) is \(2\sqrt{10}\approx6.32\) units.
The perimeter of quadrilateral \(KLMN\) is \(\sqrt{73}+\sqrt{34}+5 + 2\sqrt{10}\approx25.69\) units.