Pythagoras' Theorem (2D)
Master Pythagoras' theorem for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-04). In a right-angled triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides, so if you know any two sides you can find the third.
You will use Pythagoras' theorem on real-life problems drawn on flat diagrams — finding the hypotenuse or a shorter side, sketching a diagram first and rounding with the correct units — for everyday situations such as ladders against walls, screen diagonals and walking east then north.
Theory
Pythagoras' theorem says that in a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides: \(c^2=a^2+b^2\). In Year 12 Mathematics Standard 1 (NSW) you use it to find the hypotenuse or a shorter side when two sides are known, in problems such as ladders against walls and screen diagonals.
Pythagoras' theorem: in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides, so if you know any two sides you can find the third.
The hypotenuse \(c\) is the side opposite the right angle; it is always the longest side. The other two sides \(a\) and \(b\) are the shorter sides (legs). Named by letters, the hypotenuse joins the two vertices that are not the right angle, e.g. if the right angle is at \(K\) in \(\triangle JKL\), the hypotenuse is \(JL\).
To find the hypotenuse, square the two shorter sides and add, then take the square root. To find a shorter side, square the hypotenuse and subtract the square of the known side. In practical problems (ladders and walls, diagonals of rectangles, walking east then north) draw the right-angled triangle first, round as asked and include units.
Pythagoras' theorem, with \(c\) the hypotenuse:
Finding the hypotenuse (add the squares):
Finding a shorter side (subtract the squares):
How to use Pythagoras' theorem
- Sketch the right-angled triangle and identify the hypotenuse: the side opposite the right angle.
- Decide: finding the hypotenuse means add the squares; finding a shorter side means subtract.
- Substitute into \(c^2=a^2+b^2\) and solve: take the square root, then round as asked and include units.
Sketch the triangle: the shorter sides are \(7\) cm and \(24\) cm, so add their squares.
| \(x^2\) | \(=\) | \(7^2+24^2\) |
| \(x^2\) | \(=\) | \(49+576\) |
| \(x^2\) | \(=\) | \(625\) |
| \(x\) | \(=\) | \(\sqrt{625}=25\) |
\(x=\) \(25\) cm.
Sketch the triangle: the hypotenuse (\(26\) m) is known, so subtract the square of the other side.
| \(26^2\) | \(=\) | \(10^2+x^2\) |
| \(676\) | \(=\) | \(100+x^2\) |
| \(x^2\) | \(=\) | \(676-100=576\) |
| \(x\) | \(=\) | \(\sqrt{576}=24\) |
\(x=\) \(24\) m.
Sketch the triangle: the ladder is the hypotenuse, so subtract.
| \(4.2^2\) | \(=\) | \(1.2^2+h^2\) |
| \(17.64\) | \(=\) | \(1.44+h^2\) |
| \(h^2\) | \(=\) | \(17.64-1.44=16.2\) |
| \(h\) | \(=\) | \(\sqrt{16.2}=4.0249\ldots\) |
\(h\approx\) \(4.02\) m.
Sketch the screen: the diagonal is the hypotenuse of a right-angled triangle, so add.
| \(d^2\) | \(=\) | \(110^2+62^2\) |
| \(d^2\) | \(=\) | \(12100+3844=15944\) |
| \(d\) | \(=\) | \(\sqrt{15944}=126.269\ldots\) |
\(d\approx\) \(126.27\) cm.
Common pitfalls
Frequently asked questions
What is Pythagoras' theorem?
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: c squared equals a squared plus b squared, where c is the hypotenuse.
How do you find the hypotenuse of a right-angled triangle?
Square the two shorter sides, add them, then take the square root. For sides 7 cm and 24 cm, 49 plus 576 is 625, and the square root of 625 is 25 cm.
How do you find a shorter side using Pythagoras' theorem?
Square the hypotenuse, subtract the square of the known shorter side, then take the square root. For a hypotenuse of 26 m and a side of 10 m, 676 minus 100 is 576, so the side is 24 m.
Which side is the hypotenuse?
The hypotenuse is the side opposite the right angle. It is always the longest side of a right-angled triangle.
How do you use Pythagoras' theorem for a ladder against a wall?
The ladder is the hypotenuse, the wall and the ground are the shorter sides. For a 4.2 m ladder with its foot 1.2 m from the wall, the height is the square root of 4.2 squared minus 1.2 squared, about 4.02 m.