We are given two diagonally opposite vertices of a square on a coordinate grid, with coordinates $(-4, 7)$ and $(2, 1)$. We need to find the perimeter of the square.

1. Problem Description

We are given two diagonally opposite vertices of a square on a coordinate grid, with coordinates $(-4, 7)$ and $(2, 1)$. We need to find the perimeter of the square.

2. Solution Steps

First, we find the length of the diagonal of the square, which is the distance between the two given points. The distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

In our case, $(x_1, y_1) = (-4, 7)$ and $(x_2, y_2) = (2, 1)$.

So, the length of the diagonal $d$ is:

$d = \sqrt{(2 - (-4))^2 + (1 - 7)^2} = \sqrt{(2+4)^2 + (1-7)^2} = \sqrt{6^2 + (-6)^2} = \sqrt{36 + 36} = \sqrt{72} = 6\sqrt{2}$

Let $s$ be the side length of the square. Then, by the Pythagorean theorem, the diagonal is related to the side length by:

$d = s\sqrt{2}$

So, $s\sqrt{2} = 6\sqrt{2}$

$s = 6$

The perimeter $P$ of the square is given by:

$P = 4s$

$P = 4(6) = 24$

3. Final Answer

The perimeter of the square is 24.