Hero Triangles 

The Background:

Hero’s formula for the area of any triangle with sides a, b, c  is 

                            A  = ,

 where s is the semi-perimeter    1 (a + b + c)

 If a, b, c and A are all whole numbers, we call the triangle a Hero Triangle. 

We shall also refer to a right-angled triangle  whose sides are all whole numbers as a Pythagoras Triangle.

Why is it obvious that all Pythagoras triangles are Hero triangles?  (Think ‘Area?’)

Let’s take the Pythagoras triangles (3,4,5) and (8,15,17), and enlarge the first to (6,8,10). Now we can place them together, using the common height 8 to form two Hero triangles:             

                          

 which give Hero triangles (10,17,21) and (10,17, 9).  Check their areas by the formula and by using the areas of the two Pythagoras triangles.

There will of course be isosceles Hero triangles, made from two congruent Pythagoras triangles, such as (5,5,6) and (5,5,8).  Again, check their areas by the formula.

 The Problem :

A         Produce at least a dozen Hero triangles by this method. You will probably find it helpful to have a list of ‘Pythagoras’ triangles at the ready...   Check the Hero’s areas.

 B        Some Hero triangles are given below. Can you split each of  them up into the two original Pythagoras triangles that were put together to make them ?

Some simple Hero Triangles

 Isosceles:        5,5,6      5,5,8        17,17,16           17,17,30           25,25,14         65,65,120

 Scalene:        13,15,14      13,15,4      10,17,21     10,17,9      13,20,21       17,25,12       29,25,36

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Last modified: June 18, 2007