To prove a very important relationship between the lengths of the sides of right triangles exists.
Before you begin this activity, define each of the terms listed below. Be sure that when you are discussing this investigation with your classmates, you use these words correctly in context.
The study of mathematics is like a huge pyramid, with things which are proven forming the base of more complicated work above it. The great thing about math is that once someone has proven something, you can take it as completely true and never have to doubt its validity, and so you can use them without worrying whether they are really true or not. Unlike science, where things which are widely believed can occasionally be proven wrong, in mathematics once something has been proven properly, it is perfectly and completely true forever.
The activity we will engage in, below, will prove a very important relationship exists between the lengths of the three sides in ANY right triangle.
What the activity above has shown is that the relationship below is true for all right triangles where the side lengths $a$a and $b$b represent the legs of a right triangle and $c$c represents the hypotenuse of the right triangle. This special relationship between the lengths of the three sides of any right triangle is called Pythagorean Theorem.
Another way to verify that Pythagorean Theorem ($a^2+b^2=c^2$a2+b2=c2 ) for any right triangle, where $c$c is the hypotenuse and the shorter side lengths are $a$a and $b$b is by manipulating the applet below.
Can we manipulate this picture to make Pythagorean theorem make sense graphically?
Write in words what you think is happening from the display the Geogebra applet gives. Then, discuss your answer with a partner.
Consider the image below. At the center is a right triangle with sides measuring $a$a, $b$b and $c$c. Each of the three squares has a side length equivalent to the side length of the triangle.
This means that:
According to the Pythagorean Theorem, the area of the two smaller squares adds up to the area of the largest square. That is:
Now, if we take away the squares, this leaves us with a relationship between the side lengths of any right triangle.
Which of the two activities showed why Pythagorean Theorem is always true the most clearly for you? Discuss with a partner why you felt the representation in the activity you selected made the most sense to you.
Use inductive reasoning to explain the pythagorean relationship.