What is the side length of a square with a diagonal of 10 cm?

For a square with side length $s$ and diagonal $d$, the Pythagorean theorem states:

$$s^2 + s^2 = d^2$$ $$2s^2 = d^2$$

Solving for $s$:

$$s^2 = \frac{d^2}{2}$$ $$s = \sqrt{\frac{d^2}{2}}$$ $$s = \frac{d}{\sqrt{2}}$$

The diagonal length $d = 10$ cm.

$$s = \frac{10}{\sqrt{2}}$$

To rationalize the denominator, multiply the numerator and denominator by $\sqrt{2}$:

$$s = \frac{10}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}$$ $$s = \frac{10\sqrt{2}}{2}$$ $$s = 5\sqrt{2}$$

The side length of the square is $5\sqrt{2}$ cm.

$$s = 5\sqrt{2} \text{ cm}$$ $$s \approx 5 \times 1.4142 \text{ cm}$$ $$s \approx 7.071 \text{ cm}$$

The side length of the square is $5\sqrt{2} \text{ cm}$.