Two Crossing Ladders

Date: 07/01/2003 at 10:52:40
From: Bob Paterick
Subject: Two crossing ladders

Two walls are 10 ft. apart. Two ladders, one 15 ft. long and one 20 
ft. long, are placed at the bottoms of the walls leaning against the 
opposite walls. How far from the ground is the point of intersection?

Date: 07/02/2003 at 11:32:06
From: Doctor Greenie
Subject: Re: Two crossing ladders

Hi, Bob -

Here is an approach that uses only the idea of similar triangles.

  B
  *
  * *
  *   *
  *     *
  *       *                     D
  *         *                   *
  *           *              *  *
  *             *         *     *
  *               * E  *        *
  *                 *           *
  *              *  * *         *
  *           *     *   *       *
  *        *        *     *     *
  *     *           *       *   *
  *  *              *         * *
  *******************************
  A                 F           C

We have

   AC = 10; AD = 15; BC = 20

We can find AB and CD using the Pythagorean Theorem.

We want to find EF.

Triangles AEF and ADC are similar, so

  EF/CD = AE/AD

or

  EF = (CD)*(AE/AD)

We know CD and AD, so we will be done if we can determine AE.

To find AE, note that triangles ABE and DCE are also similar, so

  AE/DE = AB/CD

or

  AE/(15-AE) = AB/CD

We know AB and CD, so we can determine AE from this equation.  Then, 
knowing AE, we can determine EF.

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/