The perimeter of the polygon to the nearest tenth of a unit \(\boxed{13.6{\text{ units}}}\). Option (d) is correct.
Further explanation:
The distance between the two points can be calculated as follows,
\(\boxed{{\text{Distance}} = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} }\)
Given:
The coordinates of the vertices of the polygon are \(\left( { -2, 7} \right),\left( {2, 5} \right),\left( {- 1, 5} \right)\), and \(\left( { - 1, 3} \right).\)
Explanation:
The coordinates of the vertices of the polygon are \(\left( { -2, 7} \right),\left( {2, 5} \right),\left( {- 1, 5} \right)\), and \(\left( { - 1, 3} \right).\)
The distance between points \(\left( { -2, 7} \right)\) and \(\left( {2, 5} \right),\) can be calculated as follows,
\(\begin{aligned}{\text{Distance}} &= \sqrt {{{\left( { - 2 - 2} \right)}^2} + {{\left( {5 - 7} \right)}^2}}\\&= \sqrt {16 + 4}\\&= \sqrt {20}\\&= 4.47\\\end{aligned}\)
The distance between points \(\left( {2, 5} \right)\) and \(\left( {- 1, 5} \right)\) can be calculated as follows,
\(\begin{aligned}{\text{Distance}}&= \sqrt {{{\left( { - 1 - 2} \right)}^2} + {{\left( {5 - 5} \right)}^2}}\\&= \sqrt {9 + 0}\\&= \sqrt9\\&= 3\\\end{aligned}\)
The distance between points \(\left( {- 1, 5} \right)\), and \(\left( { - 1, 3} \right)\) can be calculated as follows,
\(\begin{aligned}{\text{Distance}} &= \sqrt {{{\left( { - 1 + 1} \right)}^2} + {{\left( {3 - 5} \right)}^2}}\\&= \sqrt {0 + 4}\\&= \sqrt4\\&= 2\\\end{aligned}\)
The distance between point \(\left( { -2, 7} \right)\) and \(\left( { - 1, 3} \right)\) can be calculated as follows,
\(\begin{aligned}{\text{Distance}} &= \sqrt {{{\left( { - 1 + 2} \right)}^2} + {{\left( {3 - 7} \right)}^2}} \\&= \sqrt {1 + 16} \\&= \sqrt {17} \\&= 4.13\\\end{aligned}\)
The perimeter of the polygon can be calculated as follows,
\(\begin{aligned}{\text{Perimeter}} &= 4.47 + 3 + 2 + 4.13\\&= 13.6\\\end{aligned}\)
Option (a) is not correct.
Option (b) is not correct.
Option (c) is not correct.
The perimeter of the polygon to the nearest tenth of a unit \(\boxed{13.6{\text{ units}}}.\) Option (d) is correct.
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Coordinate geometry
Keywords: perimeter, shape, perimeter of shape below, Coordinates, vertices, polygon, x-coordinate, y-coordinate, perimeter, circumference, nearest tenth unit, distance formula.
