The Pythagorean Theorem

Ontario 09 Applied (MFM1P)

What is the Pythagorean Theorem? (Investigation)

Lesson

The Pythagorean Theorem states that $a^2+b^2=c^2$`a`2+`b`2=`c`2 for any right-angled triangle, where $c$`c` is the hypotenuse and the shorter side lengths are $a$`a` and $b$`b`.

But can we picture this graphically?

Write in words what you think is happening from the display the geogebra applet gives.

Now let's look at the explanation and see how close you were!

Consider the image below. At the centre is a right-angled 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-angled triangle*.

Relate the geometric representation of the Pythagorean theorem to the algebraic representation a^2 + b^2 = c^2