To find the sum of the lengths of the two diagonals of a square when given that one side (denoted $YH$) has a length of 14 units, we can follow these steps:

Step 1: Relationship between the side and the diagonal of a square

For a square, the length of the diagonal $d$ is related to the side length $s$ by the Pythagorean theorem. Since both sides of the square are equal, the diagonal forms a right triangle with two sides of the square as the legs.

The formula for the diagonal $d$ of a square is:

$d = s \sqrt{2}$

Where $s$ is the side length of the square.

Step 2: Calculate the length of the diagonal

We are given that the side length of the square, $YH$, is 14 units. Substituting $s = 14$ into the formula for the diagonal:

$d = 14 \sqrt{2}$

Now, calculate the diagonal length approximately:

$d \approx 14 \times 1.414 \approx 19.8 \, \text{units}$

Step 3: Sum of the two diagonals

A square has two diagonals, and since both diagonals are equal in length, the sum of the lengths of the two diagonals is:

$\text{Sum of diagonals} = 2d = 2 \times 14 \sqrt{2} \approx 2 \times 19.8 = 39.6 \, \text{units}$

Conclusion

The sum of the lengths of the two diagonals is approximately 39.6 units.

Would you like a visual drawing of this?