Pythagoras’ Theorem

Pythagoras' Theorem

The Pythagoras' theorem is used to find the length of the sides of a right-angled triangle. The theorem states that, "the square of the length of the hypotenuse is equal to the sum of the squares of the length of base and height". That is, the Pythagoras' theorem says, $a^2 + b^2 = c^2$. The theorem is formed by using the right-angled triangle with lengths of the base $a$, height $b$, and hypotenuse $c$ as shown.

Verification of the Pythagoras' theorem

One of the methods used to verify the Pythagoras' theorem is to use the area of squares formed by the sides of the right-angled triangle. The theorem can be verified by counting unit squares for each square formed by the base, height, and hypotenuse. For example, use the right-angled triangle with the base of 4 centimetres, height of 3 centimetres, and hypotenuse of 5 centimetres. Follow the following steps to verify the Pythagoras' theorem:

1. Draw the right-angled triangle. Draw with a base of 4 centimetres, height of 3 centimetres, and hypotenuse of 5 centimetres.
2. Make unit squares. Make unit squares from the base, height and hypotenuse as shown.
3. Count the base unit squares. Count the number of unit squares resulting from the base of 4 centimetres. The total number of unit squares is 16.
4. Count the height unit squares. Count the number of unit squares resulting from the height of 3 centimetres. The total number of unit squares is 9.
5. Count the hypotenuse unit squares. Count the number of unit squares resulting from the hypotenuse of 5 centimetres. The total number of unit squares is 25.
6. Find the sum. Find the total number of unit squares of the base and height. The total is $16 + 9 = 25$.
7. Compare the answers. Compare the answer in steps 5 and 6. The answers are the same and the total number of unit squares is 25.

Therefore, the Pythagoras' theorem states that:

Thus, $16 + 9 = 25$ unit squares. But, $16 = 4^2$, $9 = 3^2$ and $25 = 5^2$. Therefore, $16 + 9 = 25$ unit squares can be written as: $(4\ \text{cm})^2 + (3\ \text{cm})^2 = (5\ \text{cm})^2$, where 4 cm is the length of base, 3 cm is the length of height, and 5 cm is the length of hypotenuse. If $a$ is the length of the base, $b$ is the length of the height and $c$ is the length of the hypotenuse, then $a^2 + b^2 = c^2$. Therefore, $a^2 + b^2 = c^2$, which is the Pythagoras' theorem.

Calculation of lengths of the hypotenuse, height, and base of the right-angled triangle by using the Pythagoras' theorem

Two sides of the right-angled triangle can be used to calculate the third side by using the Pythagoras' theorem. The following examples illustrate the use of the Pythagoras' theorem in determining the length of one side of the right-angled triangle when the lengths of the two other sides are given.

Calculation of length of the hypotenuse

When the lengths of the base and height of the right-angled triangle are given, then you can find the length of the hypotenuse. If the length of the base is $a$, length of the height is $b$, and length of hypotenuse is $c$, then apply the Pythagoras' theorem $a^2 + b^2 = c^2$ to calculate the length of the hypotenuse as follows: Find the square root of each side as follows: $\sqrt{a^2 + b^2} = \sqrt{c^2}$ $\sqrt{a^2 + b^2} = c$ $c = \sqrt{a^2 + b^2}$

Therefore, the length of the hypotenuse is equal to the square root of the sum of the squares of the lengths of base and height.

Example 1

Find the value of $y$ in the following right-angled triangle:

Solution

Using the Pythagoras' theorem: $a^2 + b^2 = c^2$ Base = $a$, Height = $b$ and Hypotenuse = $c$. From the figure: $c = y$, $a = 8$ cm and $b = 15$ cm. Thus, $y^2 = (8\ \text{cm})^2 + (15\ \text{cm})^2$ $y^2 = 64\ \text{cm}^2 + 225\ \text{cm}^2$ $y^2 = 289\ \text{cm}^2$ Find the square root on both sides: $\sqrt{y^2} = \sqrt{289\ \text{cm}^2}$ $y = 17$ cm Therefore, the value of $y$ is 17 cm.

Example 2

Find the value of $z$ in the following right-angled triangle:

Solution

Using the Pythagoras' theorem: $a^2 + b^2 = c^2$ Base = $a$, Height = $b$ and Hypotenuse = $c$. From the figure, $a = 12$ cm, $b = 9$ cm and $c = z$. Thus, $z^2 = (12\ \text{cm})^2 + (9\ \text{cm})^2$ $z^2 = 144\ \text{cm}^2 + 81\ \text{cm}^2$ $z^2 = 225\ \text{cm}^2$ Find the square root on both sides: $\sqrt{z^2} = \sqrt{225\ \text{cm}^2}$ $z = 15$ cm Therefore, the value of $z$ is 15 cm.

Example 3

Find the value of $k$ in the following right-angled triangle:

Solution

Using the Pythagoras' theorem: $a^2 + b^2 = c^2$ Base = $a$, Height = $b$ and Hypotenuse = $c$. From the figure, $a = 16$ cm, $b = 12$ cm and $c = k$. Thus, $k^2 = (16\ \text{cm})^2 + (12\ \text{cm})^2$ $= 256\ \text{cm}^2 + 144\ \text{cm}^2$ $= 400\ \text{cm}^2$ Find the square root on both sides: $\sqrt{k^2} = \sqrt{400\ \text{cm}^2}$ $k = 20$ cm Therefore, the value of $k$ is 20 cm.