Let's work through this problem step-by-step to determine the perimeter of the square given its diagonally opposite vertices (-2, 9) and (5, 2).
### Step 1: Calculate the Length of the Diagonal
1. Identify the coordinates of the points:
- [tex]\((x_1, y_1) = (-2, 9)\)[/tex]
- [tex]\((x_2, y_2) = (5, 2)\)[/tex]
2. Use the distance formula to find the length of the diagonal:
The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
\text{diagonal length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
3. Substitute the coordinates into the formula:
[tex]\[
\text{diagonal length} = \sqrt{(5 - (-2))^2 + (2 - 9)^2}
\][/tex]
Simplify the expressions inside the square root:
[tex]\[
\text{diagonal length} = \sqrt{(5 + 2)^2 + (2 - 9)^2}
\][/tex]
[tex]\[
= \sqrt{7^2 + (-7)^2}
\][/tex]
[tex]\[
= \sqrt{49 + 49}
\][/tex]
[tex]\[
= \sqrt{98}
\][/tex]
[tex]\[
= 9.899494936611665
\][/tex]
### Step 2: Calculate the Length of One Side of the Square
1. Relationship between diagonal and side:
The diagonal of a square relates to its side length by a 45°-45°-90° triangle. Specifically:
[tex]\[
\text{diagonal} = \text{side} \times \sqrt{2}
\][/tex]
2. Re-arrange the formula to solve for the side length:
[tex]\[
\text{side length} = \frac{\text{diagonal}}{\sqrt{2}}
\][/tex]
3. Substitute the length of the diagonal:
[tex]\[
\text{side length} = \frac{9.899494936611665}{\sqrt{2}}
\][/tex]
[tex]\[
= 6.999999999999999
\][/tex]
### Step 3: Calculate the Perimeter of the Square
1. Formula for the perimeter of a square:
[tex]\[
\text{perimeter} = 4 \times \text{side length}
\][/tex]
2. Substitute the side length:
[tex]\[
\text{perimeter} = 4 \times 6.999999999999999
\][/tex]
[tex]\[
= 27.999999999999996
\][/tex]
Hence, the perimeter of the square is [tex]\( 27.999999999999996 \)[/tex].
