Question

find the perimeter of the polygon pqr shown in the graph.
a) 6+\sqrt{29}+\sqrt{41} units
b) 6\sqrt{1189} units
c) 4\sqrt{13} units
d) 6 units

Explanation:

Step1: Find the length of \(PQ\)

Points \(P(-4,1)\) and \(Q(2,1)\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), here \(y_1 = y_2 = 1\), so \(PQ=\sqrt{(2-(-4))^2+(1 - 1)^2}=\sqrt{(6)^2+0}=6\)

Step2: Find the length of \(PR\)

Points \(P(-4,1)\) and \(R(0,6)\). \(PR=\sqrt{(0-(-4))^2+(6 - 1)^2}=\sqrt{4^2+5^2}=\sqrt{16 + 25}=\sqrt{41}\)

Step3: Find the length of \(QR\)

Points \(Q(2,1)\) and \(R(0,6)\). \(QR=\sqrt{(0 - 2)^2+(6 - 1)^2}=\sqrt{(-2)^2+5^2}=\sqrt{4+25}=\sqrt{29}\)

Step4: Calculate the perimeter

Perimeter \(=PQ+PR+QR\). Substitute the values: \(6+\sqrt{41}+\sqrt{29}\)

Answer:

A. \(6+\sqrt{29}+\sqrt{41}\) units