We have a right square pyramid with a base side of 10 cm and a height of 12 cm. We need to find the length of a non-base edge, which is one of the edges connecting the apex to the vertices of the square base.

1. Problem Description

We have a right square pyramid with a base side of 10 cm and a height of 12 cm. We need to find the length of a non-base edge, which is one of the edges connecting the apex to the vertices of the square base.

2. Solution Steps

Let the side of the square base be $a$, and the height of the pyramid be $h$. In our case, $a = 10$ cm and $h = 12$ cm.

Let the length of the non-base edge be $l$.

The non-base edge connects the apex to each vertex of the square base. Let's consider the triangle formed by the height of the pyramid, half of the diagonal of the square base, and the non-base edge. This is a right triangle.

The diagonal of the square base is $d = a\sqrt{2} = 10\sqrt{2}$.

Half of the diagonal is $d/2 = \frac{10\sqrt{2}}{2} = 5\sqrt{2}$.

Now, we use the Pythagorean theorem:

$l^2 = h^2 + (d/2)^2$

$l^2 = h^2 + (\frac{a\sqrt{2}}{2})^2$

$l^2 = h^2 + \frac{2a^2}{4}$

$l^2 = h^2 + \frac{a^2}{2}$

$l^2 = 12^2 + \frac{10^2}{2}$

$l^2 = 144 + \frac{100}{2}$

$l^2 = 144 + 50$

$l^2 = 194$

$l = \sqrt{194}$

3. Final Answer

$\sqrt{194}$