Explaining Pythagorean Relationships with Inductive Reasoning

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Pythagorean relationships are used throughout math, science and engineering. In this lesson, we use inductive reasoning to explain why these relationships are true.

Pythagorean Relationships and Inductive Reasoning

Inductive reasoning is part of everyday life. For example, Joanna and Sonia are twin sisters. We know Joanna likes to stay fit by working out at the gym using an exercise bike. We also know that Sonia likes to stay fit as well. We might conclude that Sonia also works out at the gym. Given what we know, and using some intuition, this is a reasonable conclusion.

This type of reasoning is also used in math explanations.

In this lesson we use inductive reasoning to explain Pythagorean relationships. This style of reasoning uses a mix of basic math facts with intuitively acceptable arguments to arrive at conclusions. ''Intuition'' is the vague feeling when you know something is obviously true even when it can't be completely explained -- like intuitively knowing this will be a good day.

Intuitively, I suspect Sonia does math problems while on the exercise bike...

The Goal

The goal in this lesson is to explain a2 + b2 = c2. This is the summary statement of Pythagorean relationships for the three sides of a right triangle. Refer to the following a-b-c- right triangle:

The a-b-c-right triangle
defining_triangle

The hypotenuse c (the side opposite the right angle), the shorter leg b and the longer leg a form a right triangle. (For the special case of equal legs, make ''a'' equal to ''b'', and this lesson will be fine). A right triangle is a triangle which has 90o as one of its three angles.

Our first example will use observations about two sketched drawings and the area of squares formula from geometry. Another observation is exercise bikes and math reasoning probably don't do well together at the same time.

A Five-Sided Figure and the Area of Squares

Let's construct a five-sided figure with lengths a + b, a + b, a, c and b.

A five-sided figure
five-sided_figure

Let's construct a second, identical five-sided figure, rotate it, and place it alongside the first one:

Two identical five-sided figures
two_five-sided_figures

Without even calculating, it's clear the two figures have the same area. Intuitively, if we subtract the same amount of area from each figure, each will be left with less area, but those leftover areas will still be equal.

There are 3 a-b-c right triangles in each figure. Do you see them?

Six a-b-c-triangles
six_a-b-c_triangles_in_the_figures

The three triangles labelled 1, 2 and 3 in the first figure are the same as the three triangles labelled 4, 5 and 6 in the second figure. Removing triangles 1, 2 and 3 from the first figure will reduce the area of the first figure. Similarly, removing triangles 4, 5 and 6 from the second figure will reduce the area of the second figure. The leftover area in the first figure will equal the leftover area in the second figure. Removing the triangles gives:

The remaining areas
remaining_areas

These leftover areas are squares. By referring back to a previous drawing, you can verify the labels a, b and c.

Use the area of squares formula: A = s2 where A is the area and ''s'' is the length of a side. The two squares remaining in the first figure have sides ''a'' and ''b''. Thus, the leftover area from the first figure is a2 + b2. For the second figure, the square has a side of length ''c''. The leftover area from the second figure is c2. Since the leftover areas are equal, we have explained (inductively): a2 + b2 = c2.

The next example uses ideas from similar triangles.

A gym exercise similar to the exercise bike and the rowing machine is the treadmill. I wonder if the two sisters use this similar machine in their workout.

Forming a Rectangle from Similar Triangles

If a triangle is a scaled version of another triangle, we have similar triangles.

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