Geometrical Problem: What is the Value of DC?

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    Geometrical Geometry
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Discussion Overview

The discussion revolves around a geometrical problem involving two intersecting ladders and the determination of the length DC. Participants explore various methods to approach the problem, including numerical and algebraic solutions, while seeking a closed form solution.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents the problem and asks for the value of DC, providing the lengths of the ladders and the height of the intersection point O.
  • Another participant expresses that they have an answer but seeks confirmation or assistance from others.
  • Some participants mention having found numerical solutions and express interest in deriving a closed form solution.
  • A participant describes their approach involving setting up five unknowns and using the Pythagorean theorem and similar triangles to create equations.
  • There is mention of ending up with a system of nonlinear equations, specifically a quartic equation, which some participants acknowledge as a common outcome.
  • Participants express curiosity about alternative methods that might yield a more elegant solution.

Areas of Agreement / Disagreement

Participants generally agree on the complexity of the problem and the resulting quartic equation, but there is no consensus on the best approach or solution method. Multiple competing views on how to tackle the problem remain present.

Contextual Notes

Some participants note the challenge of finding a closed form solution and the complexity introduced by the quartic equations, indicating that the problem may have multiple layers of difficulty depending on the chosen method.

n7imo
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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.
problem.png
 
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Well, I have an answer. Do you know the answer, or are you asking for help?
 
phyzguy said:
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.
 
Mathematica gave me a closed form solution, but it is very messy.
 
phyzguy said:
Mathematica gave me a closed form solution, but it is very messy.
How did you approach the problem ?
 
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.
 
phyzguy said:
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
 
Yes, I agree that you end up with a quartic.
 
phyzguy said:
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.
 

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