1. Stating the problem: We are given a square with diagonals of length 10 cm and need to find the length of each side of the square. 2. Recall that in a square, the diagonals are equal in length and each diagonal divides the square into two right-angled triangles. 3. Using the Pythagorean theorem, if the side length is $s$, then the diagonal length $d$ satisfies $$d = s\sqrt{2}$$ because the diagonal is the hypotenuse of a right triangle with legs of length $s$. 4. Given $d = 10$ cm, we solve for $s$: $$10 = s\sqrt{2}$$ 5. Divide both sides by $\sqrt{2}$ to isolate $s$: $$s = \frac{10}{\sqrt{2}}$$ 6. Rationalize the denominator: $$s = \frac{10}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{10\sqrt{2}}{2} = 5\sqrt{2}$$ 7. Approximate the value: $$5\sqrt{2} \approx 5 \times 1.414 = 7.07$$ cm Answer: The side length of the square is $5\sqrt{2}$ cm or approximately 7.07 cm.