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### Overlapping right triangle problem

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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

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
```
Associated Topics:
College Triangles and Other Polygons
High School Triangles and Other Polygons

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