The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.

Use the Pythagorean Theorem to find the length of the diagonal.

$\displaystyle \text{Diagonal}=\sqrt{\text{side length}^2+\text{side length}^2}$

$\displaystyle \text{Diagonal}=\sqrt{2(\text{side length})^2}$

$\displaystyle \text{Diagonal}=(\text{side length})\sqrt2$

$\displaystyle \text{Side length}=\frac{\text{Diagonal}}{\sqrt2}$

For the square given in the question,

$\displaystyle \text{Side length}=\frac{9\sqrt2}{\sqrt{2}}$

Simplify.

$\displaystyle \text{Side length}=9$

The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.

Use the Pythagorean Theorem to find the length of the diagonal.

$\displaystyle \text{Diagonal}=\sqrt{\text{side length}^2+\text{side length}^2}$

$\displaystyle \text{Diagonal}=\sqrt{2(\text{side length})^2}$

$\displaystyle \text{Diagonal}=(\text{side length})\sqrt2$

$\displaystyle \text{Side length}=\frac{\text{Diagonal}}{\sqrt2}$

For the square given in the question,

$\displaystyle \text{Side length}=\frac{88\sqrt2}{\sqrt{2}}$

Simplify.

$\displaystyle \text{Side length}=88$

The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.

Use the Pythagorean Theorem to find the length of the diagonal.

$\displaystyle \text{Diagonal}=\sqrt{\text{side length}^2+\text{side length}^2}$

$\displaystyle \text{Diagonal}=\sqrt{2(\text{side length})^2}$

$\displaystyle \text{Diagonal}=(\text{side length})\sqrt2$

$\displaystyle \text{Side length}=\frac{\text{Diagonal}}{\sqrt2}$

For the square given in the question,

$\displaystyle \text{Side length}=\frac{97\sqrt2}{\sqrt{2}}$

Simplify.

$\displaystyle \text{Side length}=97$

The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.

Use the Pythagorean Theorem to find the length of the diagonal.

$\displaystyle \text{Diagonal}=\sqrt{\text{side length}^2+\text{side length}^2}$

$\displaystyle \text{Diagonal}=\sqrt{2(\text{side length})^2}$

$\displaystyle \text{Diagonal}=(\text{side length})\sqrt2$

$\displaystyle \text{Side length}=\frac{\text{Diagonal}}{\sqrt2}$

For the square given in the question,

$\displaystyle \text{Side length}=\frac{12}{\sqrt{2}}$

Multiply the top and bottom of the fraction by one in the form of $\displaystyle \frac{\sqrt{2}}{\sqrt{2}}$

$\displaystyle \text{Side length}=\frac{12}{\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}}$

Solve.

$\displaystyle \text{Side length}=\frac{12\sqrt2}{2}$

Simplify.

$\displaystyle \text{Side length}=6\sqrt2$

The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.

Use the Pythagorean Theorem to find the length of the diagonal.

$\displaystyle \text{Diagonal}=\sqrt{\text{side length}^2+\text{side length}^2}$

$\displaystyle \text{Diagonal}=\sqrt{2(\text{side length})^2}$

$\displaystyle \text{Diagonal}=(\text{side length})\sqrt2$

$\displaystyle \text{Side length}=\frac{\text{Diagonal}}{\sqrt2}$

For the square given in the question,

$\displaystyle \text{Side length}=\frac{10}{\sqrt{2}}=\frac{10\sqrt2}{2}=5\sqrt2$

Multiply the top and bottom of the fraction by one in the form of $\displaystyle \frac{\sqrt{2}}{\sqrt{2}}$

$\displaystyle \text{Side length}=\frac{10}{\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}}$

Silve.

$\displaystyle \text{Side length}=\frac{10\sqrt2}{2}$

Simplify.

$\displaystyle \text{Side length}=5\sqrt2$

To find the answer to this question, the formula for area is needed. 

$\displaystyle Area = Side(Side)$

Plugging in the numbers, we get:

$\displaystyle 16 = Side (Side)$

This is also written as 

$\displaystyle 16 = Side^2$

Take the sqare root of 16, and you get:

$\displaystyle 4cm = side$

To solve, simply use the formula for the area of a square to solve for side length. Thus,

$\displaystyle A=s^2\Rightarrow s=\sqrt{A}$

$\displaystyle s=\sqrt{144}=12$

Since both sides of a square are of equal length, we can use this formula to determine the length of both sides.

$\displaystyle \\Area=length\cdot length\\ 729\;ft^2=length^2\\length=\sqrt{729\;ft^2}\\length=27\;ft$

The volume of a cube is found by multiplying the width, height, and depth. Since all sides are equal, the formula would look like $\displaystyle V=s^3$. Since we know the volume, we can take the cube root of that volume to get the individual side length.

$\displaystyle 512=s^3$

What number cubed equals 512? 8 cubed gives 512.

To solve, set up an equation using the Pythagorean Theorem:

$\displaystyle x^2 + x^2 = \sqrt{8}^2$

$\displaystyle x^2 + x^2 = 8$

$\displaystyle 2x^2 = 8$ divide both sides by 2

$\displaystyle x^2 = 4$ take the square root of both sides

$\displaystyle x = 2$