The perimeters of two squares are 52 meters and 326 meters. Find the length of the side of the square whose area is equal to the sum of the areas of the two squares.

Steps to Solve

1. Calculate the side lengths of the given squares

Let's denote the perimeters of the two squares as $P_1$ and $P_2$. The side length of a square can be found using the formula:

$$ s = \frac{P}{4} $$

So, the side lengths $s_1$ and $s_2$ can be calculated as:

$$ s_1 = \frac{P_1}{4} $$

$$ s_2 = \frac{P_2}{4} $$

2. Find the areas of the two squares

Once we have the side lengths, we can find the areas $A_1$ and $A_2$ of the squares using the formula:

$$ A = s^2 $$

The areas will be:

$$ A_1 = s_1^2 $$

$$ A_2 = s_2^2 $$

3. Sum the areas of the two squares

Now, we need to calculate the total area $A_{total}$ of both squares:

$$ A_{total} = A_1 + A_2 = s_1^2 + s_2^2 $$

4. Determine the side length of the new square

Now that we have the total area, we find the side length $s_{new}$ of the new square whose area is equal to $A_{total}$:

$$ s_{new} = \sqrt{A_{total}} $$

So we can write:

$$ s_{new} = \sqrt{s_1^2 + s_2^2} $$