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 \(PQ\)

The coordinates of \(P(-4,1)\) and \(Q(2,1)\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), since \(y_1 = y_2 = 1\), \(d_{PQ}=\sqrt{(2-(-4))^2+(1 - 1)^2}=\sqrt{(6)^2+0}=6\)

Step2: Find the length of \(PR\)

The coordinates of \(P(-4,1)\) and \(R(0,6)\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), \(d_{PR}=\sqrt{(0 - (-4))^2+(6 - 1)^2}=\sqrt{16 + 25}=\sqrt{41}\)

Step3: Find the length of \(QR\)

The coordinates of \(Q(2,1)\) and \(R(0,6)\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), \(d_{QR}=\sqrt{(0 - 2)^2+(6 - 1)^2}=\sqrt{4+25}=\sqrt{29}\)

Step4: Calculate the perimeter

The perimeter \(P=d_{PQ}+d_{PR}+d_{QR}\), so \(P = 6+\sqrt{41}+\sqrt{29}\)

Answer:

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