Pythagorean Theorem
The Pythagorean Theorem describes an important geometric relationship between the three sides of a Right Triangle. A Right Triangle is a triangle with a \(90^{\circ}\) angle (indicated by an 'L' shape). The longest side is called the hypotenuse which lies across from the \(90^{\circ}\) angle.
A Right Triangle has the following rule:
The square of the hypotenuse is equal to the sum of the squares of the other two sides.
By drawing a square on each side of the triangle, the equation says that the area of the largest square on the hypotenuse is equal to the sum of the area of the other squares.
The Pythagorean Theorem equation is:
\(\textcolor{blue}{a^2} + \textcolor{red}{b^2} = \textcolor{purple}{c^2}\)
where \(c\) is the hypotenuse and \(a\) and \(b\) are the other sides called the opposite and adjacent sides (order does not matter).
Integer values of \(a, b,\) and \(c\) that satisfy the equation are known as Pythagorean Triples. Some well known examples are \((3,4,5)\) and \((5,12,13)\). We can multiply the entries in a triple by any integer and get another triple such as \((3,4,5) * 2 = (6,8,10), (3,4,5) * 3 = (9,12,15), (3,4,5) * 5 = (15,20,25)\), etc.
Pythagorean Theorem Proof
The Pythagorean Theorem can also be proven simply with a diagram and a couple equations.
Refer to the diagram below. The big square with the side length of \(a + b\) has encased a smaller square with length \(c\). The remaining portions in the big square are triangles with lengths \(a\) and \(b\). Notice that these are right triangles!
First, we can represent the area of the big square as such:
\(A_{\text{Big Square}} = (a + b)(a + b)\)
Next, we can represent the area of the small square as such:
\(A_{\text{Small Square}} = c^2\)
Then, we can represent the area of each of the four triangles as such:
\(A_{\text{Triangle}} = \cfrac{ab}{2}\)
Notice that the area of the big square is equal to the area of smaller one plus 4 triangles. We can represent this as:
\(A_{\text{Big Square}} = c^2 + 4\left(\cfrac{ab}{2}\right)\)
\(A_{\text{Big Square}} = c^2 + 2ab\)
After, we can set the two equations for the big square equal to each other:
\((a + b)(a + b) = c^2 + 2ab\)
Finally, we can expand and simplify the left side of the equation:
\(a^2 + 2ab + b^2 = c^2 + 2ab\)
\(a^2 + b^2 = c^2\)
This is the Pythagorean Theorem!
We can use the Pythagorean Theorem to calculate the missing side of a triangle. Try a couple of examples below!
Example
In a right angle triangle, the adjacent side is \(20 \; [\text{km}]\) and opposite side is \(15 \; [\text{km}]\). What is the length of the triangle's hypotenuse?
We can use Pythagorean Theorem to solve for the missing side. In this instance, we will set \(\textcolor{blue}{a = 20}\) and \(\textcolor{red}{b = 15}\) to solve for \(\textcolor{purple}{c}\):
\(\textcolor{purple}{c}^2 = \textcolor{blue}{a}^2 + \textcolor{red}{b}^2\)
\(\textcolor{purple}{c}^2 = (\textcolor{blue}{20})^2 + (\textcolor{red}{15})^2\)
\(c^2 = 400 + 225\)
\(c^2 = 625\)
\(\sqrt{c^2} = \sqrt{625}\)
\(c = 25 \; [\text{km}]\)
Therefore, we can determine that the length of the hypotenuse is \(\boldsymbol{25 \; [\textbf{km}]}\).
First, we can draw a sketch of the square to help visualize the problem:
Adding a diagonal creates two triangles. We know that the corners of a sqare are right angles which makes this a right triangle. The missing side is the diagonal which is the hypotenuse (largest side and across from the right angle).
Next, since we can identify \(\textcolor{blue}{a = 1}\) and \(\textcolor{red}{b = 1}\), we can use the Pythagorean Theorem to solve for \(\textcolor{purple}{c}\):
\(\textcolor{purple}{c}^2 = \textcolor{blue}{a}^2 + \textcolor{red}{b}^2\)
\(\textcolor{purple}{c}^2 = \textcolor{blue}{1}^2 + \textcolor{red}{1}^2\)
\(c^2 = 1 + 1\)
\(c^2 = 2\)
\(\sqrt{c^2} = \sqrt{2}\)
\(c = 1.41\)
Therefore, we can determine that the length of the diagonal is \(\boldsymbol{1.41}\).
To check if this is a right angle triangle we will check if it satisfies the Pythagorean Theorem.
The largest side is \(\textcolor{purple}{c=26}\) (hypotenuse). The other sides are the opposite and adjacent (\(a,b\) ). In this instance, we will set \(\textcolor{blue}{a = 24}\) and \(\textcolor{red}{b = 10}\):
\(\textcolor{purple}{c}^2 = \textcolor{blue}{a}^2 + \textcolor{red}{b}^2\)
\(\textcolor{purple}{26}^2 = \textcolor{blue}{24}^2 + \textcolor{red}{10}^2\)
\(676 = 576 + 100\)
\(676 = 676\)
Therefore, since both sides are equal, we can determine that this triangle is a Right Triangle.
We can find how high the ladder will reach by picturing the scenerio as a diagram. We notice that the ladder, wall and ground make a triangle. Since the ground is flat and the wall goes straight up, they make a \(90^{\circ}\) angle, making this a right triangle!
First, we can sketch a diagram to help visualize the problem:
We can solve for value of the opposite side using the Pythagorean theorem. In this instance, we will set \(\textcolor{blue}{a = 12}\) and \(\textcolor{purple}{c = 15}\) to solve for \(\textcolor{red}{b}\):
\(\textcolor{purple}{c}^2 = \textcolor{blue}{a}^2 + \textcolor{red}{b}^2\)
\(\textcolor{purple}{15}^2 = \textcolor{blue}{12}^2 + \textcolor{red}{b}^2\)
\(b^ 2 = 225 - 144\)
\(b^ 2 = 81\)
\(\sqrt{b^2} = \sqrt{81}\)
\(b = 9 \; [\text{ft}]\)
Therefore, we can determine that the ladder will reach \(\boldsymbol{9 \; [\textbf{ft}]}\) up the wall.
Area of a Right Triangle
The Area of a Right Triangle can be calculated using the following formula:
\(A = \cfrac{1}{2}bh\)
The area is calculated using the lengths of the two shorter sides as the base, \(b\), and the height, \(h\).
If one of these dimensions is unknown and you know the hypoteneuse, apply the Pythagorean theorem to calculate the length of the unknown side. Then, use the area formula.
Example
Calculate the area of the following right triangle:
First, we need to determine the height length. In order to do so, we can use the Pythagorean theorem:
\(\textcolor{blue}{a}^2 + \textcolor{red}{b}^2 = \textcolor{purple}{c}^2\)
\((\textcolor{blue}{5})^2 + \textcolor{red}{b}^2 = (\textcolor{purple}{8})^2\)
\(25 + b^2 = 64\)
\(b^2 = 64 - 25\)
\(b^2 = 39\)
\(\sqrt{b^2} = \sqrt{39}\)
\(b \approx 2.45 \; [\text{cm}]\)
Next, we can determine the area of triangle by applying its corresponding formula:
\(A = \cfrac{1}{2}bh\)
\(A = \cfrac{1}{2}(5)(6.25)\)
\(A \approx 16.125\; [\text{cm}^2]\)
Therefore, we can determine that the area of the right triangle is approximately \(\boldsymbol{16.125\; [\textbf{cm}^2]}\).
First, we need to determine base length. In order to do so, we can use the Pythagorean theorem:
\(\textcolor{blue}{a}^2 + \textcolor{red}{b}^2 = \textcolor{purple}{c}^2\)
\((\textcolor{blue}{a})^2 + \textcolor{red}{12}^2 = (\textcolor{purple}{15})^2\)
\(144 + b^2 = 225\)
\(b^2 = 225 - 144\)
\(b^2 = 81\)
\(\sqrt{b^2} = \sqrt{81}\)
\(b = 9 \; [\text{m}]\)
Next, we can determine the area of triangle by applying its corresponding formula:
\(A = \cfrac{1}{2}bh\)
\(A = \cfrac{1}{2}(9)(12)\)
\(A = 54\; [\text{m}^2]\)
Therefore, we can determine that the area of the right triangle is \(\boldsymbol{54\; [\textbf{m}^2]}\).