It’s Professor Dave, I wanna tell you about the Pythagorean Theorem.
There is one more very important theorem we need to discuss regarding triangles.
While some theorems seem pretty obvious, there is a theorem relating the side lengths of
a right triangle that is not so obvious.
This theorem is called the Pythagorean theorem, named for Pythagoras himself, even though
the theorem was likely discovered prior to Ancient Greece.
Nevertheless it was an impressive feat of logic, and one of the most famous mathematical
relationships, so let’s go over it now.
In a right triangle, there is one ninety degree angle, by definition.
The two sides that meet to form this right angle are called the legs of the triangle,
and the third side, the one opposite the right angle, is called the hypotenuse.
This theorem states that the sum of the squares of the legs of a right triangle is equal to
the square of the hypotenuse.
This is represented by the equation A squared plus B squared equals C squared, where A and
B are the legs, and C is the hypotenuse.
This means that given any two sides of a right triangle, we can calculate the third.
For example, let’s say the legs have lengths of five and twelve.
We just plug these in as A and B, square them, and we get twenty-five plus one hundred forty
four, which is one hundred sixty nine.
Since this is equal to C squared, we have to take the square root of both sides.
This gives us plus or minus thirteen for C. Since this is not algebra but rather geometry,
and we are looking at shapes, we can throw away the negative value, because it doesn’t
make sense for a side length to be negative, so we can definitively say that this hypotenuse
has a length of thirteen.
Likewise, we could know the hypotenuse and one leg, and calculate the other leg.
Say we have five and nine.
That means five squared plus B squared equals nine squared.
Twenty-five plus B squared equals eighty-one, B squared equals fifty-six, and B must be
root fifty six, which can be simplified to root four times root fourteen, or two root
fourteen.
The lengths won’t always be neat and tidy integers, so we’d better get used to this.
Sometimes we do algebra using the pythagorean theorem.
Say the legs of a right triangle are equal to X and four root three, while the hypotenuse
is equal to two X.
What are all the side lengths?
Well we can plug these terms into the theorem just like we would any numbers.
After squaring everything, on the left we have X squared, plus four squared is sixteen
times root three squared which is three, giving us forty-eight.
On the right, two X quantity squared is four X squared.
We subtract X squared, divide by three, and take the square root.
Once again, although we technically get plus and minus four, negative four doesn’t make
any sense here, so X must be four.
That makes the missing sides four and eight.
This theorem is incredibly useful and ubiquitous in math.
So how was it derived?
How can we know for sure that this relationship between the sides of a right triangle is true?
A satisfactory proof will require the ability to calculate the area of certain shapes, so
I promise we will return to this question very soon.
For now, let’s check comprehension.