Find the perimeter of the triangle whose vertices are $(-2,-1),(1,-1),$ and $(1,3)$ . Write the exact answer. Do not round. $\newline$ Answer

Full solution

Q. Find the perimeter of the triangle whose vertices are

(-2,-1),(1,-1),

and

(1,3)

. Write the exact answer. Do not round.

\newline

Answer

Calculate Distance Between Vertices: Calculate the distance between the first two vertices $(-2,-1)$ and $(1,-1)$ . Use the distance formula: $d = \sqrt{((x_2 - x_1)^2 + (y_2 - y_1)^2)}$ . Here, $(x_1, y_1) = (-2, -1)$ and $(x_2, y_2) = (1, -1)$ . $d = \sqrt{((1 - (-2))^2 + (-1 - (-1))^2)} = \sqrt{((1 + 2)^2 + (0)^2)} = \sqrt{(3^2 + 0)} = \sqrt{9} = 3$ .
Calculate Distance Between Vertices: Calculate the distance between the second and third vertices $(1,-1)$ and $(1,3)$ . Use the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ . Here, $(x_1, y_1) = (1, -1)$ and $(x_2, y_2) = (1, 3)$ . $d = \sqrt{(1 - 1)^2 + (3 - (-1))^2} = \sqrt{(0)^2 + (3 + 1)^2} = \sqrt{0 + 4^2} = \sqrt{16} = 4$ .
Calculate Distance Between Vertices: Calculate the distance between the third vertex $(1,3)$ and the first vertex $(-2,-1)$ . Use the distance formula: $d = \sqrt{((x_2 - x_1)^2 + (y_2 - y_1)^2)}$ . Here, $(x_1, y_1) = (1, 3)$ and $(x_2, y_2) = (-2, -1)$ . $d = \sqrt{((-2 - 1)^2 + (-1 - 3)^2)} = \sqrt{((-3)^2 + (-4)^2)} = \sqrt{(9 + 16)} = \sqrt{25} = 5$ .
Find Perimeter of Triangle: Add the distances from steps $1$ , $2$ , and $3$ to find the perimeter of the triangle. $\newline$ Perimeter = $\text{side}_1 + \text{side}_2 + \text{side}_3 = 3 + 4 + 5 = 12$ .