Can a Geometric Mean be Nice?

Part Two

Now that we have seen how to find the geometric mean and use it in proportions. Let's see how this relates to the pythagorean theorem. We can actually use the geometric mean proportions to "discover" the Pythagorean theorem.

Remember that h is the altitude of the largest triangle. Also remember that the proportions exist;

 c = a a x
which gives us a^2 = c * x

 c = b b y
which gives us b^2 = c * y

 Remember: The Pythagorean Theorem is a^2 + b^2 = c^2 in this triangle. and a^2 = c * x b^2 = c * y

Using some general algebraic manipulations we find that;

• a^2 + b^2 = c*x + c * y
• a^2 + b^2 = c(x+y)
• a^2 + b^2 = c(c)
• a^2 + b^2 = c^2

Hence, we arrive at the Pythagorean Theorem.

This method of demonstrating the Theorem of Pythagoras is attributed to Bhaskara a Hindu Mathematician around 1150AD.

Having Fun? Good! Go to the.......

Interactive Java Sketch

or Back to the First Page or experiment with the Pythagorean Theorem below.

Calculating the Hypotenuse of a Right Triangle
When Given Two Leg Lengths

 Legs Lengths A = B =
or
 Hypotenuse (C)= A^2+B^2 Area = (1/2)*A*B