How Many Congruent Triangles?

Date: 11/11/2001 at 15:07:37
From: Sally Gregor
Subject: Geometry

Suppose you are given a scalene triangle and a point P on some line L. 
How many triangles are there with one vertex at P, another vertex on 
L, and each triangle congruent to the given triangle? 

I used graph paper and found that is might be four. Is that right?

Date: 11/12/2001 at 06:55:47
From: Doctor Floor
Subject: Re: Geometry

Dear Sally,

Thanks for writing.

The answer to your question depends on the distance from P to L. 

If that distance is, for instance, a mile, and the given triangle has 
sides of 3, 4 and 5 inches, then clearly there is not a triangle that 
satisfies your description.

If on the other hand the distance between P and L is 1 inch, so less 
than the lengths of the three given triangle sides, then there are 
TWELVE possibilities; for each side length - a, b and c - there are 
four.

Let's say you choose sidelength a. You can find the four triangles as 
follows:

  1. From P you can draw the circle with a radius of three inches; 
     this circle meets L in two points, say Q1 and Q2.

  2. Now PQ1 is one side of the triangle. You can find a third vertex, 
     say R1, that fits to PQ1 to form a triangle congruent to the 
     given one, in such a way that PQ1R1 are oriented clockwise.

  3. On the other side of PQ1 there is a second point R2 that also 
     yields a triangle PQ1R2 congruent to the given one, but now 
     oriented counter-clockwise.

  4. With Q2 instead of Q1, steps 2 and 3 can be repeated, so that we 
     again find two triangles congruent to the given one.

Instead of PQ1 and PQ2 being of length a, you can also choose them to 
be of lengths b and c. That gives the total of 12 possibilities.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/