The Pythagorean Theorem

By Curtis Bustos

The Pythagorean Theorem was named after a Greek mathematician and philosopher named Pythagoras. While Pythagoras has been credited with many mathematical contributions, the Pythagorean Theorem a² + b² = c² is one of his most notable and has made enormous contributions toward the study of trigonometry.

The Pythagorean Theorem states that if you have a right triangle, the square of the hypotenuse is equal to the square of the other two sides of that triangle also known as a² + b² = c². The hypotenuse is the long side of a right-angled triangle and is represented by the letter “c” while the other two sides or legs  “a & b” are each shorter relative to the hypotenuse. The hypotenuse can also be defined as the side opposite to the right angle (90˚ ) of a triangle.

Learning Goal – how the Pythagorean Theorem can be applied to right-angles and their corresponding triangles.

Learning Activity – 5E Instructional model using think-pair-share.

5E Instructional model:

1.     Engagement: draw the students in with a reading.

2.     Exploration: focus on concepts and skills. Mathematical examples are provided.

i.       Introduce a right triangle.  The Pythagorean Theorem is only valid with a right triangle (90˚).  Explain obtuse and acute triangles for comparability.

ii.     Explain terms in relation to a, b, c

A = (legs or right-angle sides)

B = (legs or right-angle sides)

C = Hypotenuse (longest side and opposite to angle). 

a² +b² = c². This theorem states that the sum of the squares of the two right angle sides will always be the same as the hypotenuse.

a² + b² = c² (right angle)

a² + b² < c² (obtuse angle)

a² + b² > c² (acute angle)

iii.    Figure out a² + b² = c² where

                           

and

 

3.     Explanation: Builds on the first two phases by allowing students to determine if they are working with a right triangle?

Use a triangle with 3’, 4’, 5’.

4.     Elaboration: real world application.  Pythagorean theorem is important to drafters, engineers, carpenters, astronomers, x-ray crystallography and many additional important trades. Without x-ray crystallography, our understanding of biological processes would be reduced to a much lower level than we consider standard today. When scientists measure the directions and intensities of x-ray beams, they can deduce chemical structures and biological processes. This would not have been attainable without the famous Pythagorean Theorem.

5.     Evaluation (student understanding):

 What is the diagonal distance across a square?