Hey, this is Presh Talwalkar, and you're watching Mind Your Decisions.
Oxford ranks as one of the top universities in the world.
A version of this problem was used as an Oxford interview question.
A ladder is placed vertically on a wall.
What shape does the midpoint of the ladder trace as the ladder slides against the wall to the floor?
What if the ladder instead fell backwards to the ground by rotating from its base?
What shape would the midpoint of the ladder trace then?
Can you figure it out?
Give this problem a try, and when you're ready keep watching the video for a solution.
So, how can we solve this problem?
Well, it sometimes helps in problems like these to sketch out a few cases and get some intuition of what the answer might be.
So let's place this ladder vertically on the wall and consider the path of its midpoint.
We can imagine the ladder sliding against the wall and we'll keep track of where the midpoint of this ladder goes.
Eventually, we'll get some idea of what this path is.
Now focusing just on the midpoints we might guess that this shape is a quarter circle
But Oxford interviews - just like my videos - are about more than getting just the correct answer.
It's about demonstrating your ability to think and to prove the answer.
So can we prove that the path is a quarter circle?
Well, in order to do that, let's go back to our sketch and consider one position of the ladder.
Let's write L for the length of the ladder.
Above and below the midpoint will be L/2 because we have half the ladders length in each direction.
Now consider these two right triangles.
Because they have the same angles and their hypotenuse has the same length these two right triangles are congruent to each other.
So they will have X as their horizontal length and Y as the vertical length
Now the midpoint will be (x, y) in a coordinate system.
Now looking at either triangle,
we can use the Pythagorean Theorem to get x² + y² = (L/2)².
This equation is very well-known. In this case the midpoint traces out a quarter circle in the positive quadrant
with the radius that's half the ladder's length. And this is the proof of why the path is a quarter circle.
But there's nothing quite like seeing it for yourself
So here's an animation that I made on Desmos.com, and you can see that the midpoint of the ladder
exactly traces out the path a quarter-circle as it slides against the wall.
So now let's consider the second problem,
which is the ladder falling away from the wall.
Like before let's sketch each of these cases and get an idea for the path of the midpoint.
Now looking at this path we might once again guess that it's a quarter circle.
But can we prove it?
Well, in this case it's a little bit easier:
The midpoint will always be a distance of L/2 from the corner of the wall in the ground.
So you can imagine that's the same distance and therefore it's a quarter circle.
But we can also prove it like in the first case where we use the Pythagorean Theorem
and we get exactly the same equations as before to show that this midpoint also traces out
the path of a quarter circle.
So once again, seeing is also satisfying so here's an animation that's made with Desmos.com to show
that the midpoint traces out the path of a quarter circle as the ladder falls away from the wall.
And we now have a rather curious result:
Both cases show the same path of the midpoint!
So here's another animation that puts both ladders in
the same graph and you'll see the midpoint of each ladder
is along this quarter circle.
So amazingly, both cases result in the same path of a quarter circle.
Now obviously in real life the sliding ladder will be slower due to its friction against the wall in the ground.
This animation is just to show why the paths of the midpoints are the same shape.
A geometric way that you can see this is that the two ladders represent the diagonals of a rectangle,
and they will intersect at this midpoint of the two ladders.
You can also see this as the pivot point of a scissor lift.
So it's quite amazing
that in either case the midpoint of the ladder is tracing out the same quarter circle
Did you figure it out?
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