Question

find the perimeter and area of the figure if each unit on the graph measures 1 centimeter. round answers to the nearest tenth, if necessary

Explanation:

Step1: Use the distance formula

The distance formula is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For side \(a\) (between \((- 2,3)\) and \((2,-1)\)):
\(a=\sqrt{(2 + 2)^2+(-1 - 3)^2}=\sqrt{16 + 16}=\sqrt{32}\approx5.7\)
For side \(b\) (between \((2,-1)\) and \((5,4)\)):
\(b=\sqrt{(5 - 2)^2+(4 + 1)^2}=\sqrt{9 + 25}=\sqrt{34}\approx5.8\)
For side \(c\) (between \((5,4)\) and \((-2,3)\)):
\(c=\sqrt{(5 + 2)^2+(4 - 3)^2}=\sqrt{49+1}=\sqrt{50}\approx7.1\)
Perimeter \(P=a + b + c\approx5.7+5.8 + 7.1=18.6\)

Step2: Use Heron's formula

Let \(s=\frac{a + b + c}{2}=\frac{5.7+5.8+7.1}{2}=9.3\)
Area \(A=\sqrt{s(s - a)(s - b)(s - c)}\)
\(A=\sqrt{9.3(9.3 - 5.7)(9.3 - 5.8)(9.3 - 7.1)}\)
\(A=\sqrt{9.3\times3.6\times3.5\times2.2}\)
\(A=\sqrt{255.084}\approx16.0\)

Answer:

Perimeter: \(18.6\) centimeters, Area: \(16.0\) square centimeters