A very well known theorem is the Pythagorean theorem, founded by Pythagoras. This theorem explains the relationship among the three sides of a right angle triangle.
It states that the square of the hypotenuse, the side opposite to the right angle, is equal to the sum of the squares of the two other sides connected to the right angle.
This theorem can be written as an equation where \(a\), \(b\), and \(c\) are the sides of the triangle. This equation is called the Pythagorean equation, it can be written as shown below,
\[a^2 + b^2 = c^2.\]
In this case, \(c\) represents the hypotenuse of the triangle, and \(a\) & \(b\) represent the two other sides of the triangle.
We can use this equation/formula to find the length of any side if the other two side lengths are provided. Let’s find out how through an example.
Let’s say we have a triangle as shown below:
We can highlight the hypotenuse (the side opposite to the right angle) to make it easier to view:
Now, we can tell the side lengths of the two legs very easily, but not the hypotenuse. We know that the height \((a)\) is \(5\) units, and the width \((b)\) is \(3\) units.
If we substitute these values for \(a\) and \(b\) in the Pythagorean equation, we should be able to find the length of the hypotenuse fairly quickly. We should get,
\[a^2 + b^2 = c^2,\] \[5^2 + 3^2 = c^2,\] \[25 + 9 = c^2,\] \[34 = c^2,\] \[\sqrt{34} = c.\]
Since \(\sqrt{34}\) is pretty messy \((5.830951…)\), we’ll just leave it as \(\sqrt{34}\).
Thus, we’ve now got \(c = \sqrt{34}\) which means that the hypotenuse is \(\sqrt{34}\).
We’ve now shown how we can use the Pythagorean theorem to find the hypotenuse, or any side of a right angle triangle.