Question

what is the perimeter of a polygon with vertices at $(-1,3)$, $(-1,6)$, $(2,10)$, $(5,6)$, and $(5,3)$?
enter your answer in the box. do not round any side lengths.

units

Explanation:

Step1: Calculate length of vertical and horizontal sides

For vertical sides (same \(x\)-coordinate):

• Between \((-1,3)\) and \((-1,6)\): \(|6 - 3|=3\)
• Between \((5,3)\) and \((5,6)\): \(|6 - 3| = 3\)

For horizontal sides (same \(y\)-coordinate):

• Between \((-1,3)\) and \((5,3)\): \(|5-(-1)|=6\)
• Between \((-1,6)\) and \((5,6)\): \(|5 - (-1)|=6\)

Step2: Calculate length of slanted side

Use distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) for \((-1,6)\) and \((2,10)\)
\(d=\sqrt{(2-(-1))^2+(10 - 6)^2}=\sqrt{3^2+4^2}=\sqrt{9 + 16}=\sqrt{25}=5\)
Also, for \((2,10)\) and \((5,6)\): \(d=\sqrt{(5 - 2)^2+(6 - 10)^2}=\sqrt{3^2+(-4)^2}=\sqrt{9+16}=\sqrt{25}=5\)

Step3: Sum up all side lengths

Perimeter \(P=3 + 6+5+6+3=23\)

Answer:

\(23\)