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

### Rotate the Square

```Date: 09/19/2002 at 13:41:21
Subject: Geometry Construction

I have the following construction that I've been trying to figure out.

First I've connected A to P and A to Q. Then I've found the midpoint
of each line segment and constructed the half circles (k and l)
outside the square, with AP and AQ as their respective diameters.
I know that B and D must lie on k and l respectively because they are
guaranteed to have 90-degree angles, by Thales' theorem: if any
line segment (RS) is the diameter of a circle, any point on the
circle (T) makes a 90-degree angle when connected to the ends of the
line segment (angle RTS = 90 degrees).

The question is: Which points on the half-circles are B and D?

My teacher mentioned something about transforming the circle l so that
it intersects with k. Some of my classmates and I cannot figure out
the transformations. Can you help?  Or do you know of another way to
construct the square?
```

```
Date: 09/19/2002 at 17:31:07
From: Doctor Floor
Subject: Re: Geometry Construction

Hi, Gabrielle,

Let me summarize the question: we have points APQ and must construct
a square ABCD such that P is on BC and Q is on CD.

I have done this problem before, and I always use a very simple
construction. It is based on the following:

1. If we rotate square ABCD together with point Q about A through
-90 degrees (clockwise) we will get a square attached to AB,
and the image of Q, say Q', will lie on the line through B and C.

2. In the same way if we rotate ABCD together with point P about A
through 90 degrees (counterclockwise) we will get a square attached
to AD and the image P' of P will lie on the line through D and C.

So the construction is as follows:

Rotate P about A through 90 degrees, to find P'.
Rotate Q about A through -90 degrees, to find Q'.
Line P'Q contains D and C.
Line Q'P contains B and C.
So the intersection of P'Q and Q'P gives C.

Having A and C, and the lines through C in the right directions, the
rest should be easy.

If you have more questions, just write back.

Best regards,
- Doctor Floor, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Constructions

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