Consider the following situation:
One could think of the two diagonal line segments as ladders leaning against two opposite walls of an alley. Two interesting questions may be asked:
Given the heights of the two ladders above the ground, how high above the ground is the point where they meet?
If the lengths of the ladders are given, as well as the height of their intersection point, how wide is the alley?
Let's explore some basic cases:
- If , how much is
- If i.e. is much smaller than how much is
If , the situation is perfectly symmetric. The intersection point lies precisely halfway between the two walls. Because the intersection point lies halfway on each ladder, its height is also half of each ladder's height:
If , the intersection point lies very close to the left wall. Its height is only slightly less than the height of the first ladder:
If the ladders have the same length and cross at height , what is the width of the alley?
In this case, we are again dealing with the symmetric situation. From the discussion above we know that . Applying the Pythagorean theorem, we find
The answer to question 1 is
Define a coordinate system with the -axis along the ground. Without loss of generality, the vertical walls have equations and . The two ladders can be described by the following linear equations:
Equating the two, we obtain
and for the vertical coordinate we find