Lesson - #7 Pythagorean Theorem and Triples

Initial lesson posted - July, 2011
Revision date - Fall 2017

Purpose: This lesson introduces the Pythagorean Theorem. A related concept is the Pythagorean Triple.
Essential Questions: What is the Pythagorean Theorem and how can it be used to find a missing side length in right triangles?

A Section of the Lesson Follows

The Pythagorean Theorem states that if you are given a right triangle, the sum of the squares of the legs’ lengths will equal the square of the length of the hypotenuse.
This theorem is one of the most powerful theorems in mathematics because it allows us to calculate a missing side length in any right triangle. Each of the shorter sides of a triangle are called legs and the longest side is called the hypotenuse. The proof of the theorem is beyond the scope of this series of lessons.
Solve for missing side lengths:
All rectangles can be decomposed into two right triangles. The consecutive sides (do not call them adjacent sides) are the legs and the diagonal (connect the opposite vertices) is the hypotenuse. The hypotenuse is shared between the two right triangles. The reason that this works is because all of the angles in a rectangle are right angles.
Note that in this lesson, I used the setxy command in the coordinate plane to draw the diagonal (hypotenuse) in the Logo rectangles.
Example 1:
Figure 7.1 shows rectangle ABCD with dimensions 3 units by 4 units. Figure 7.2 shows a rectangle created in Logo using the dimensions 30 by 40 pixels. A scale factor of ten units was used to create the Logo triangle because 3 turtle steps would not show up on the screen.
The setxy command connects the point (0,0) to the point (40, 30).
The essential question is, what is the length of the diagonal (segment DB) in these rectangles?

Omitted Section of Lesson #7

Example 4:
What if we know the measures of the hypotenuse and one of the legs in a given right triangle? Can we work backwards to find the missing leg length? Yes! Find the missing side lengths in these examples.


Omitted Section of Lesson #7

Pythagorean Triples
The converse of the Pythagorean theorem states that if the sum of the squares of the two shorter sides of a given triangle equals the square of the longest side, then the triangle is a right triangle.
A Pythagorean Triple is a set of three positive integer values that satisfy the converse of the Pythagorean theorem. We will begin our discussion with the triple 3-4-5. Note the positive integers are the set of numbers {1, 2, 3, … }.
The 3-4-5 triple is the one that most people recognize, because if you square the two smaller values (the 3 and the 4) and add the squares, you get 9 +16, which is 25.
The calculation confirms that IF the sides are 3-4-5 THEN the triangle is a right triangle.
Since 3, 4 or 5 turtle steps are very small in Logo, you can use a similar triangle with larger dimensions. The common factor in 30, 40, and 50 is ten. Multiply each side length for the 3-4-5 triangle by ten and you get 30-40-50 as the new side lengths (Example 1).
The result of using a scale factor of 100 with the 3-4-5 triple, would be a similar triangle with side dimensions 300, 400 and 500. The calculation below verifies the relationship.
300^2 + 400^2 = 500^2 or 90000 + 160000 = 250000