To find the perimeter of the triangle with vertices [tex]\((-2, -5)\)[/tex], [tex]\((3, -5)\)[/tex], and [tex]\((3, 7)\)[/tex], we can follow these steps:
1. Determine the lengths of the sides of the triangle using the distance formula. The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
2. Calculate the length of the side between [tex]\((-2, -5)\)[/tex] and [tex]\((3, -5)\)[/tex]. These points have the same [tex]\(y\)[/tex]-coordinate, so the distance is:
[tex]\[
\text{side 1} = \sqrt{(3 - (-2))^2 + (-5 - (-5))^2} = \sqrt{(3 + 2)^2 + 0^2} = \sqrt{5^2} = 5
\][/tex]
3. Calculate the length of the side between [tex]\((3, -5)\)[/tex] and [tex]\((3, 7)\)[/tex]. These points have the same [tex]\(x\)[/tex]-coordinate, so the distance is:
[tex]\[
\text{side 2} = \sqrt{(3 - 3)^2 + (7 - (-5))^2} = \sqrt{0 + (7 + 5)^2} = \sqrt{12^2} = 12
\][/tex]
4. Calculate the length of the side between [tex]\((-2, -5)\)[/tex] and [tex]\((3, 7)\)[/tex]. The distance is:
[tex]\[
\text{side 3} = \sqrt{(3 - (-2))^2 + (7 - (-5))^2} = \sqrt{(3 + 2)^2 + (7 + 5)^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13
\][/tex]
5. Add the lengths of the three sides to find the perimeter:
[tex]\[
\text{Perimeter} = \text{side 1} + \text{side 2} + \text{side 3} = 5 + 12 + 13 = 30
\][/tex]
Therefore, the perimeter of the triangle is [tex]\(30\)[/tex].
