Overlapping right triangle problem

Date: 09/14/97 at 22:57:51
From: Judy Wilson
Subject: Overlapping right triangle problem

Given right triangles ABC and DCB with rt angles at B and C.
The hypotenuse of triangle ABC, AC, is 20 and the hypotenuse of 
triangle DCB, DB, is 30. The hypotenuses intersect at point E, which 
is a distance of 10 from BC. Find the length of BC.

I've tried similar triangles and systems of trig equations, and 
I've gotten nowhere.

Date: 09/15/97 at 14:19:56
From: Doctor Mitteldorf
Subject: Re: Overlapping right triangle problem

Dear Judy,

I drew the figure three times before I recognized this as the old 
"two ladders in an alley" problem.  One ladder is 20 feet, the other 
is 30 feet, they cross the alley in opposite directions, and they meet 
10 feet off the ground.

Here's a start: Call the width of the alley x+y, where x is the part
to the left of the ladder and y is to the right. See if we can use 
both similar triangles and Pythagoras to generate two equations in x 
and y.

The first equation, for example, says that the ratio of height to the 
base of the left "small" triangle is the same as the ratio of the 
height to the base of the "full" triangle formed by the alley, the 
right wall, and the ladder. 

   10/x = sqrt(30^2-(x+y)^2)/(x+y)

I leave it to you to write down the corresponding equation for the 
other ladder, and to solve the two equations simultaneously for 
x and y.

-Doctor Mitteldorf,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 09/15/97 at 15:57:58
From: Judy Ann Wilson
Subject: Re: Overlapping right triangle problem

Dear Dr. Mittledorf,

Thanks so much for the help. I can hardly wait to go work on this. I 
knew the problem was a classic but... 

Judy