May 10, 1999
Problem of the Month

Tricky Triangle
by Julius Cohn
in The Washington Post

The angles of a triangle are in the ratio 3:4:5.

If the shortest side has a length of 1 meter, what is the length of the longest side?

Solution to the Problem:

The answer is (1 + \/3) / 2 meters
or approximately 1.36 meters

First, determine the measures of the angles using 3x + 4x + 5x = 180 degrees.

The angles measure 45, 60, and 75 degrees respectively.

(1) You can solve the problem by basic geometry if you draw an altitude from the vertex of the 75 degree angle to the opposite base (the longest side).   This altitude divides the original triangle into two smaller triangles (a 45-45-90 triangle and a 30-60-90 triangle).   You can now determine the measures of all the sides.

(2) An alternative method is to use the Law of Sines to set up the problem.
Call the length of the short side S and the length of the longest side L.

Then sin(75) / L = sin(45) / S

Since S = 1 meter, sin (75) = .9659258 and sin(45) = .7071,
it follows that L = 1.366 meters.

Correctly solved by:

1. Jon Pence	Winchester, VA
2. Jia Ran	Rome, Italy
3. Bob Hearn	Winchester, VA

Send any comments or questions to: David Pleacher