Pythagorean Theorem
Many adults report that their high school experiences of algebra and geometry were very different from each other, such that they recall one course much more favorably than the other. Interestingly, algebra and geometry are intimately related.
In algebra when we see exponents written, such as, 52 and 53. We read the expressions as “five squared” and “five cubed.” Geometrically, these algebraic expressions can be represented as a square with side of length 5 and as a cube with side of length 5.
An excellent opportunity for a geometric representation of an algebraic expression is the Pythagorean Theorem: a2 + b2 = c2, and usually stated orally as, “The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.”
Let us consider the right triangle below with legs a and b and hypotenuse c.
Let us construct squares based on both legs and on the hypotenuse.
Now, let us reflect the square of the hypotenuse over the hypotenuse of the triangle, so that it covers parts of the squares based on the two legs.
Now, let’s construct a larger square with length of side a + b, in which these overlapping squares fit.
We can now see that the total area of the large square with side a + b is the sum of the area of the green square based on side a, the area of the blue square based on side b, and the areas of two red and yellow rectangles with dimensions a x b: (a + b)2 = a2 + b2 + 2ab.
Similarly, we can see that the total area of the large square with side a + b is the sum of the area of the red square based on side c and the areas of the yellow, green, blue, and light blue right triangles with legs a and b: (a + b)2 = c2 + 4(1/2ab) = c2 + 2ab.
Since, (a + b)2 = a2 + b2 + 2ab and (a + b)2 = c2 + 2ab,
then by substitution a2 + b2 + 2ab = c2 + 2ab. Subtracting 2ab from both sides yields a2 + b2 = c 2.
When Albert Einstein was asked for a proof of the Pythagorean Theorem, I think he must have been visualizing diagrams like those above when he responded, "What's to prove?"
