13.1 The Pythagoras Theorem
Hypotenuse
The hypotenuse is the longest side of a right-angled triangle. It is always located directly opposite the right angle ($90^\circ$).
The Pythagoras Theorem Formula
For any right-angled triangle with side lengths $a$ and $b$, and hypotenuse $c$:
$$c^2 = a^2 + b^2$$
Derived side length formulas:
- Hypotenuse ($c$): $c = \sqrt{a^2 + b^2}$
- Other sides ($a$ or $b$): $a = \sqrt{c^2 - b^2}$ and $b = \sqrt{c^2 - a^2}$
Common Pythagorean Triples
A set of three positive integers $(a, b, c)$ that satisfy $a^2 + b^2 = c^2$:
- $3, 4, 5$ (and its multiples like $6, 8, 10$ or $9, 12, 15$)
- $5, 12, 13$ (and multiples like $10, 24, 26$)
- $7, 24, 25$
- $8, 15, 17$
- $9, 40, 41$
13.2 Converse of the Pythagoras Theorem
The converse of the Pythagoras theorem is used to determine whether a given triangle is a right-angled triangle based on its side lengths.
Let $c$ be the longest side of a triangle, and $a$ and $b$ be the other two sides:
- If $c^2 = a^2 + b^2$, then the triangle is a right-angled triangle (the angle opposite side $c$ is $90^\circ$).
- If $c^2 < a^2 + b^2$, then the triangle is an acute-angled triangle (all interior angles are $< 90^\circ$).
- If $c^2 > a^2 + b^2$, then the triangle is an obtuse-angled triangle (the angle opposite side $c$ is $> 90^\circ$).