The Pythagoras' Theorem

13.1

The Pythagoras' Theorem

Hypotenuse:

Definition

The longest side of a right-angled triangle which is opposite to the right angle.

Example

Identify the hypotenuse for the following diagram.

\(AC\) is the side opposite to the right angle.

Thus, \(AC\) is the hypotenuse.

The relationship between the sides of a right-angled triangle:

The area of the square on the hypotenuse is equal to the total area of the squares on the other two sides.

\(\begin{aligned} \text{Area of }R&=\text{Area of }P+\text{Area of }Q \\\\AC^2&=AB^2+BC^2 \end{aligned}\)

This relationship is known as the Pythagoras’ theorem.

Example

State the relationship between the lengths of sides of the following right-angled triangle.

The relationship is

\(LN^2=LM^2+MN^2\).

Determine the length of the unknown side of a right-angled triangle:

Example

Given the following diagram, calculate the value of \(x\).

Give the answer in two decimal places.

\(\begin{aligned} x^2&=9^2+14^2 \\\\&=81+196 \\\\&=277 \\\\x&=\sqrt{277} \\\\&=16.64\text{ cm}. \end{aligned}\)

The lengths of the unknown side of combined geometric shapes:

Example

Calculate the length of \(QS\) in the following diagram.

First, calculate the length of \(QR\).

\(\begin{aligned} QR^2&=PQ^2-PR^2 \\\\&=15^2-12^2 \\\\&=225-144 \\\\&=81 \\\\QR&=\sqrt{81} \\\\&=9\text{ cm}. \end{aligned}\)

Next, calculate the length of \(RS\).

\(\begin{aligned} RS^2&=PS^2-PR^2 \\\\&=13^2-12^2 \\\\&=169-144 \\\\&=25 \\\\RS&=\sqrt{25} \\\\&=5\text{ cm}. \end{aligned}\)

Thus, the length of \(QS\) is

\(\begin{aligned} QS&=QR+RS \\\\&=9+5 \\\\&=14\text{ cm}. \end{aligned}\)