Geometrical Problem: What is the Value of DC?

[n7imo]

Hello,

A friend of mine gave me this puzzle and I'd like to share it with you, math enthusiasts:
Two ladders intersect in a point O, the first ladder is 3m long and the second one 2m. O is 1m from the ground, that is AC = 2, BD = 3 and OE = 1 (see the image bellow)

Question: what's the value of DC ?

Note:
E is the perpendicular projection of O on [DC] and it does not necessarily divide [DC] in two equal parts.

 

[phyzguy]

Well, I have an answer. Do you know the answer, or are you asking for help?

 

[n7imo]

Well, I have an answer. Do you know the answer, or are you asking for help?

Don't hesitate sharing it, I found the answer in a numerical form, but I'm curious about finding a closed form solution.

 

[phyzguy]

Mathematica gave me a closed form solution, but it is very messy.

 

[n7imo]

Mathematica gave me a closed form solution, but it is very messy.

How did you approach the problem ?

 

[phyzguy]

Maybe not the most elegant way, but I set up 5 unknowns - OD, OC, AD, BC, CD. Using the Pythagorean theorem and similar triangles, you can write 5 equations relating them, then solve them.

 

[n7imo]

Maybe not the most elegant way, but I set up 5 unknowns - OD, OC, AD, BC, CD. Using the Pythagorean theorem and similar triangles, you can write 5 equations relating them, then solve them.

I ended up with a system of two nonlinear equation, I talked about it here https://www.physicsforums.com/threads/a-nonlinear-equation-system.862991/
I used x =BC and y= AD. Once I found x and y I used Pythagore to find DC. But the x (or y) end up being a solution of a quartic equations ... messy

 

[phyzguy]

Yes, I agree that you end up with a quartic.

 

[n7imo]

Yes, I agree that you end up with a quartic.

The problem is very easy to explain, I wonder if there is a different approach that give an elegant solution.