A ladder placed against a wall slides down. During the sliding, the distance $A T$ from the top of the ladder to the initial rooftop position is called the "sliding down" of the ladder, and the distance $B O$ from the end point of the ladder to the corner of the wall is called the "sliding out". During the movement, which of the two quantities, the sliding down and the sliding out, is greater?

![](https://cdn.mathpix.com/cropped/2024_05_02_c35720115fbe6a238809g-1.jpg?height=474&width=330&top_left_y=316&top_left_x=884)
We will show that while the ladder is in motion - that is, it has already started moving but has not yet completely slipped - the slipping is always greater than the sliding.

![](https://cdn.mathpix.com/cropped/2024_05_02_b917ab8cc05c8b84311fg-1.jpg?height=469&width=339&top_left_y=270&top_left_x=882)

Consider the triangle \( AOB \) shown in the figure. According to the triangle inequality, \( AO + OB > AB \). On the other hand, \( AB \) is equal to the length of the ladder, so \( AB = OT = AO + TA \). Substituting this into the previous inequality:

\[
AO + OB > AO + TA, \quad \text{which means} \quad OB > TA
\]

\( OB \) is the slipping, and \( TA \) is the sliding.

Based on the work of Pétervári Erika (Tamási, Béri Balogh Á. Gymnasium, 2nd grade).

Note. Many solved the problem using the Pythagorean theorem. These solutions also received 3 points, although the problem statement does not mention that the wall is perpendicular to the ground, i.e., that \( AOB \) is a right angle. The provided solution shows that this condition is not necessary.