7. Geometry

Lesson

To prove a very important relationship between the lengths of the sides of right triangles exists.

- Paper (unlined)
- Colored pencils
- Scissors; and
- Ruler

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.

- Right triangle
- Hypotenuse
- Leg
- Area
- Area of a Square

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.

- Start with a square with four equal right triangles drawn into it, so that the hypotenuses form a square in the middle. Each of these triangles has sides $a$
`a`and $b$`b`, and hypotenuse $c$`c`. (See the diagram on the left below for specifics on how our first drawing should look. Be sure to use colored pencils to make each of the four triangles a different color, as shown ). To be sure that the four triangles we are creating are equal, it may be helpful to trace the first one and create a stencil of it . Then you can use the stencil to form the other three triangles. - How could we express the area of the white square in the middle, with its side length equal to $c$
`c`? Discuss how you arrived at the answer to this question with a partner. - Cut out the four colored triangles and rearrange the triangles as shown in the diagram on the right side. Then cut the white square that was in the middle of the first diagram so that it fits into the space available in the second diagram.
- Why do you think that the inside of the squares in the diagram on the left are labeled $a^2$
`a`2 and $b^2$`b`2? Discuss your reasoning with a partner. - Based upon your answers to questions $2$2 and $4$4 above, do you think that the equation $a^2+b^2=c^2$
`a`2+`b`2=`c`2 will always be true?

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.

Remember: Pythagorean theorem is worth memorizing, because it is used often!

Another way to verify that Pythagorean Theorem ($a^2+b^2=c^2$`a`2+`b`2=`c`2 ) 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:

- the square with side length $a$
`a`has an area of $a^2$`a`2 - the square with side length $b$
`b`has an area of $b^2$`b`2 - the square with side length $c$
`c`has an area of $c^2$`c`2

According to the Pythagorean Theorem, the area of the two smaller squares adds up to the area of the largest square. That is:

$a^2+b^2=c^2$`a`2+`b`2=`c`2

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.