Two opposite vertices of the square have coordinates (1; 3) and (-5; -3). Find the area and perimeter of the square.

Two opposite vertices of the square have coordinates (1; 3) and (-5; -3).

Find the area and perimeter of the square.

Let the vertex v1 have coordinates x1 and y1, and the vertex v2 has coordinates x2 and y2.

Then x1 = 1, y1 = 3, x2 = -5, y2 = -3.

Find the distance between these points, it will be equal to the length of the diagonal of the square.

l = √ ((x1 – x2) ^ 2 + (y1 – y2) ^ 2) = √ ((1 – (-5)) ^ 2 + (3 – (-3)) ^ 2) = √ ((1 + 5) ^ 2 + (3 + 3) ^ 2) = √6 ^ 2 + 6 ^ 2 = √72.

By the Pythagorean theorem, the length of the side of the square – x and the length of the diagonal – l have the following relation. l = √ (2 * x ^ 2) = x * √2 or x = l / √2.

Find the side length of the square: x = √72 / √2 = √ (72/2) = √36 = 6.

Then the area of ​​the square is S = x * x and the perimeter is P = 4 * x.

S = 6 * 6 = 36.

P = 4 * 6 = 24.

Answer: S = 36, P = 24.