Question

use the pythagorean theorem to help find the perimeter of the triangle.

Explanation:

Step1: Find the length of the vertical side

The vertical side has endpoints \((-2, -5)\) and \((-2, -7)\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), here \(x_1=x_2=-2\), \(y_1 = - 5\), \(y_2=-7\). So \(d=\sqrt{(-2+2)^2+(-7 + 5)^2}=\sqrt{0+(-2)^2}=2\) units.

Step2: Find the length of the horizontal side

The horizontal side has endpoints \((-2,-5)\) and \((3,-5)\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), here \(y_1=y_2=-5\), \(x_1=-2\), \(x_2 = 3\). So \(d=\sqrt{(3 + 2)^2+(-5+5)^2}=\sqrt{5^2+0}=5\) units.

Step3: Find the length of the slanted side using the Pythagorean theorem

Let \(a = 2\) (vertical side), \(b = 5\) (horizontal side). By the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), where \(c\) is the length of the slanted side. So \(c=\sqrt{2^{2}+5^{2}}=\sqrt{4 + 25}=\sqrt{29}\approx5.39\) units.

Step4: Calculate the perimeter

The perimeter \(P\) of a triangle is \(P=a + b + c\). Substituting \(a = 2\), \(b = 5\), \(c=\sqrt{29}\), we get \(P=2+5+\sqrt{29}=7+\sqrt{29}\approx7 + 5.39=12.39\) units.

Answer:

The perimeter of the triangle is \(7+\sqrt{29}\approx12.39\) units.