The Pythagorean theorem
The Pythagorean theorem relates the three sides of every right triangle. It is one of the oldest and most widely used results in mathematics.
The theorem
In a right triangle with legs a and b and hypotenuse c, the side opposite the right angle, \(a^2 + b^2 = c^2\). A triangle with legs 6 and 8 has hypotenuse \(\sqrt{36 + 64} = 10\). The hypotenuse is always the longest side.
History
The relationship is named after Pythagoras, a Greek thinker of the sixth century BCE, but it was known much earlier. A Babylonian clay tablet called Plimpton 322, from around 1800 BCE, lists sets of numbers fitting the theorem, and ancient Indian and Chinese texts also used it.
Finding a leg
To find a missing leg, subtract: if the hypotenuse is 13 and one leg is 5, the other leg is \(\sqrt{169 - 25} = \sqrt{144} = 12\). A common mistake is adding when the hypotenuse is known; always check which side is opposite the right angle.
Pythagorean triples
Whole-number solutions such as 3-4-5, 5-12-13, 8-15-17, and 7-24-25 are Pythagorean triples. Multiplying a triple by any whole number gives another: 6-8-10 and 9-12-15 come from 3-4-5. Builders have long used a 3-4-5 rope to mark right angles.
The converse
If the sides of a triangle satisfy \(a^2 + b^2 = c^2\), with c the longest, the triangle is right. If \(a^2 + b^2 > c^2\) it is acute, and if \(a^2 + b^2 < c^2\) it is obtuse. Sides 7, 8, and 11 give 49 + 64 = 113 < 121, so that triangle is obtuse.
A visual proof
Arrange four copies of a right triangle inside a square of side a + b so they leave a tilted square of side c in the middle. The big square's area is \((a + b)^2\), which also equals four triangles plus the middle square: \(4 \cdot \tfrac{1}{2}ab + c^2\). Expanding and simplifying gives \(a^2 + b^2 = c^2\).
Distance and three dimensions
The distance formula is the Pythagorean theorem on a grid. In three dimensions, a box's space diagonal is \(\sqrt{l^2 + w^2 + h^2}\): a box 3 by 4 by 12 has a diagonal of \(\sqrt{9 + 16 + 144} = 13\).
Worked example
A 5 m ladder leans against a wall with its foot 1.4 m from the wall. It reaches \(\sqrt{25 - 1.96} = \sqrt{23.04} = 4.8\) m up the wall. Moving the foot out to 3 m drops the top to \(\sqrt{25 - 9} = 4\) m. Television sizes are also quoted by diagonal: a 16 : 9 screen 88.6 cm wide is about 49.8 cm tall, giving a diagonal of about 101.6 cm, or 40 inches.
Key takeaways
In a right triangle, a squared + b squared = c squared, where c is the hypotenuse opposite the right angle. The result was known in Babylon (Plimpton 322, about 1800 BCE) long before Pythagoras. Subtract to find a leg. Pythagorean triples such as 3-4-5 and 5-12-13, and their multiples, give whole-number sides. The converse classifies triangles as right, acute, or obtuse by comparing a squared + b squared with c squared. The theorem gives the distance formula and the space diagonal of a box.
Flashcard
Side opposite the right angle
Check your understanding
A right triangle has hypotenuse 13 and one leg 5. What is the other leg?