The
Binomial Expansion - Pythagorean Theorem
Problem
The
Binomial Expansion - Pythagorean Theorem
Problem
Given: In the figure above the two larger squares are congruent.
The red figure is a square. The lighter blue triangle is a right
triangle with legs a and b
and is congruent to the other triangles in the figure.
The blue square on the left is the square of the hypotenuse
of the given right triangle.
(1) Why is the area of the blue square on the left a2 + b 2?
(2) Find the total area of the 4 triangles in the square on the right.
(3) Show that the area of the red square is (b - a)2.
(4) Show that the sum of all the blue areas (light and dark) is (b + a)2.
(5) Expand the binomials (b - a)2 and (b + a)2 and add the
results together to get an algebraic expression for the total area
consisting of all parts of the figure blue (light and dark) and red.
In passing, we note that if we create a triangle whose legs are (b - a) and (b + a), then the area of the square on the hypotenuse of that triangle would be equal to the result we obtained in problem 5. Can you draw it?
