Associated Topics || Dr. Math Home || Search Dr. Math

### Right Angle

```
Date: 04/09/97 at 20:53:33
From: Raman Verma
Subject: Right Angle

I sure hope you can help me with this question. It goes like this:

B                     C
_____________________
|\                   /|
| \                /  |
|  \             /    |
|   \         /       |
|    \     /          |
|_____\/______________|
A  P          D

The rectangle above has AB = 12cm and BC = 25cm. What value(s) for AP
will make angle BPC a right angle?

I pretty sure that you have to prove the two triangles similar (I
think), but I don't know what to do after that.
```

```
Date: 04/14/97 at 18:41:55
From: Doctor Wilkinson
Subject: Re: Right Angle

You've got the right idea; you just need to run with it.

First of all, can you prove the triangles are similar?  They're both
right triangles, so you just need to show they have another pair of
corresponding angles equal.  All you have to go on is that BPC is a
right angle. This suggests looking at the angles BPA and CPD, because
these angles together with BPC make a straight angle. If you subtract
out the right angle in the middle, you find that BPA and CPD add up to
a right angle. From this you should be able to conclude that BPA and
PCD are equal. Do you see how to do that?

Now if you've got that far, then you know that triangles ABP and CPD
are similar. What do you know about similar triangles? One thing you
know is that they have corrsponding angles equal. But you already
know that. What you want to use is the fact that corresponding sides
are proportional.

So let x be the length of AP. That makes the length of PD 25 - x,
right?

So one of the two similar triangles has sides 12 and x (we're not
going to bother with the hypotenuses). The other one has sides 25 - x
and 12. Clearly the 12 doesn't correspond to the 12, so it must be the
other way around!  So we have an equation which you should be able to
solve:

x/12 = 12/(25 -x)

-Doctor Wilkinson,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Geometry
High School Triangles and Other Polygons

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search