Are derivations important?

Are derivations, especially in Geometry, important? I’ve frequently posed this question to students over the years and have typically got the response − “No, if they are out of syllabus!” I disagree almost completely. Years back when I was in school, almost inadvertently, I got down to doing something really boring on a boring summer afternoon. I picked up Heron’s formula:

Area =

Where, s is the semi-perimeter of the triangle i.e. 

I admired its unwieldy look and got down to deriving it. Quite obviously, I first drew a triangle with the sides given.

I realized pretty soon that there wasn’t much headway I could have made without doing some construction. I did what any student would do. I dropped a perpendicular and did away with my first roadblock.

Then I wrote the known formula for the area of a triangle (something I should have done first!)

Area of ΔABC

Now I had two right triangles. Right triangles ! obviously Pythagoras theorem. Suddenly I had these equations.

c² = h² + d² … (1)

b² = h² + (a − d)²

I knew that the answer had to be in terms of the lengths of the sides. I looked at the equations and did the most obvious operation to eliminate ‘h’ − I subtracted one equation from the other and the result was this − 

b² − c² = a² − 2ad

⇒ 2ad = a² + b² − c²

I got the value of ‘d’ in terms of a, b and c. I knew I was getting somewhere. I substituted this value in the first equation to write ‘h’ in terms of a, b & c.

After this I did my last step and that was simply substituting the value of ‘h’ in the formula for area. This is what I got.

Area of ΔABC

Looked scary but it worked as well as Heron’s formula (which probably was a few steps away). Now if I had a,b and c, I could find the area of the triangle!

What this did was it removed my mental block towards Heron’s formula, and that is important. In Geometry, you will almost always remember the formula/theorem but when faced with a question, you will not know which theorem to call for help when. And that is what derivations do. They give you the power over theorems, and thus Geometry. So next time if you get a theorem, attempt its derivation. You might not always be successful, but you would have made a good beginning in the direction of mastery over Geometry.

- Pavan

21 comments May 7th, 2009