The length of the diagonal of the square is 10 cm. Find the area of the square.

In a square, the length of the diagonal (d) is related to the side length (s) by the formula:

$\mathbf{d}\mathbf{=}\mathbf{s}\sqrt{\mathbf{2}}$

Given that the length of the diagonal is $10\text{ cm}$:

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

To find the side length (s), divide both sides of the equation by $\sqrt{2}$:

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

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

$\mathrm{s}=\frac{10\cdot \sqrt{2}}{2}$​​

$\mathbf{s}\mathbf{=}\mathbf{5}\sqrt{\mathbf{2}}$

Now that we know the side length (s), we can find the area (A) of the square using the formula:

$\mathbf{A}\mathbf{=}{\mathbf{s}}^{\mathbf{2}}$

$\mathrm{A}=(5\sqrt{2}{)}^2$

$\mathrm{A}=25\times 2$

$\mathbf{A}\mathbf{=}\mathbf{50}{\mathbf{\text{ cm}}}^{\mathbf{2}}$

Therefore, the area of the square is $\mathbf{50}{\mathbf{\text{ cm}}}^{\mathbf{2}}$.