1. **Stating the problem:** We have a pyramid with a vertical height of 25 cm and a base side length of 10 cm. We want to find the apothem (the slant height from the apex to the midpoint of a base side). 2. **Formula and explanation:** The apothem $a$ of a regular pyramid can be found using the Pythagorean theorem in the right triangle formed by the vertical height $h$, half the base side length $\frac{s}{2}$, and the apothem $a$ as the hypotenuse: $$a = \sqrt{h^2 + \left(\frac{s}{2}\right)^2}$$ 3. **Substitute the known values:** $$a = \sqrt{25^2 + \left(\frac{10}{2}\right)^2}$$ 4. **Simplify inside the square root:** $$a = \sqrt{625 + 5^2}$$ $$a = \sqrt{625 + 25}$$ 5. **Calculate the sum:** $$a = \sqrt{650}$$ 6. **Simplify the square root if possible:** $$a = \sqrt{25 \times 26} = 5\sqrt{26}$$ 7. **Final answer:** The apothem of the pyramid is $$a = 5\sqrt{26} \text{ cm} \approx 25.5 \text{ cm}$$