Question

what is the perimeter of the pentagon? round your answer to the nearest hundredth.

Explanation:

Step1: Calculate the length of the vertical sides

The vertical sides are from \((0,0)\) to \((0,8.5)\) and from \((17,0)\) to \((17,8.5)\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), for \((0,0)\) and \((0,8.5)\): \(d=\sqrt{(0 - 0)^2+(8.5 - 0)^2}=8.5\). Similarly, for \((17,0)\) and \((17,8.5)\), \(d = 8.5\).

Step2: Calculate the length of the two non - vertical, non - horizontal sides

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \((0,8.5)\) and \((8.5,17)\): \(d_1=\sqrt{(8.5 - 0)^2+(17 - 8.5)^2}=\sqrt{8.5^2+8.5^2}=\sqrt{2\times8.5^2}=8.5\sqrt{2}\approx12.02\).
For \((8.5,17)\) and \((17,8.5)\): \(d_2=\sqrt{(17 - 8.5)^2+(8.5 - 17)^2}=\sqrt{8.5^2+(- 8.5)^2}=8.5\sqrt{2}\approx12.02\).

Step3: Calculate the length of the horizontal side

The horizontal side is from \((0,0)\) to \((17,0)\). Using the distance formula \(d=\sqrt{(17 - 0)^2+(0 - 0)^2}=17\).

Step4: Sum up all the side lengths

Perimeter \(P=8.5 + 8.5+12.02+12.02+17\).
\(P=(8.5\times2)+(12.02\times2)+17\).
\(P = 17+24.04+17\).
\(P=58.04\).

Answer:

\(58.04\)