Pythagoras' Theorem

Introduction to Pythagoras' Theorem

Pythagoras' Theorem is one of the most famous and useful theorems in mathematics, describing a special relationship between the three sides of a right-angled triangle (a triangle containing one 90° angle). Named after the ancient Greek mathematician Pythagoras, this theorem allows us to calculate the length of any side of a right-angled triangle if the lengths of the other two sides are known, and it has countless practical applications in construction, navigation, and everyday problem-solving.

Parts of a Right-Angled Triangle

In a right-angled triangle, the side opposite the right angle (the longest side) is called the hypotenuse. The other two sides, which form the right angle itself, are simply called the legs (or sometimes the base and the height/perpendicular, depending on the orientation of the triangle). Correctly identifying the hypotenuse is the first and most important step in applying Pythagoras' Theorem correctly.

Statement of Pythagoras' Theorem

Pythagoras' Theorem states that in any right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This is written as the formula: c² = a² + b², where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides (the legs) of the triangle.

Finding the Hypotenuse

When the lengths of the two shorter sides (legs) of a right-angled triangle are known, the length of the hypotenuse can be found by rearranging the formula: c = √(a² + b²). For example, in a right-angled triangle with legs of length 3 cm and 4 cm, the hypotenuse is calculated as c = √(3² + 4²) = √(9 + 16) = √25 = 5 cm. This particular set of side lengths (3, 4, 5) is a well-known example called a Pythagorean triple.

Finding a Missing Leg

When the hypotenuse and one leg are known, the length of the missing leg can be found by rearranging the formula: a = √(c² − b²), or equivalently b = √(c² − a²). For example, in a right-angled triangle with a hypotenuse of 13 cm and one leg of 5 cm, the missing leg is found as a = √(13² − 5²) = √(169 − 25) = √144 = 12 cm.

Pythagorean Triples

A Pythagorean triple is a set of three positive whole numbers that satisfy Pythagoras' Theorem exactly, such as (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). These triples are useful to recognise and memorise, as they often appear in examination questions and allow for quick calculations without needing to work through square roots. Multiples of these triples are also valid Pythagorean triples; for example, doubling (3, 4, 5) gives (6, 8, 10), which also satisfies the theorem.

Verifying a Right-Angled Triangle

Pythagoras' Theorem can also be used in reverse, to check whether a given triangle is a right-angled triangle. If the square of the longest side equals the sum of the squares of the other two sides, then the triangle must contain a right angle. For example, to check whether a triangle with sides 6 cm, 8 cm, and 10 cm is right-angled: 10² = 100, and 6² + 8² = 36 + 64 = 100; since both values are equal, the triangle is indeed right-angled, with the right angle located opposite the 10 cm side.

The Converse of Pythagoras' Theorem

The converse of Pythagoras' Theorem states that if the square of one side of a triangle equals the sum of the squares of the other two sides, then the angle opposite that side must be a right angle. This converse is what allows the theorem to be used for verification, as described above, and is a powerful tool for confirming whether a given set of measurements forms a right angle, without needing to measure the angle directly.

Applying Pythagoras' Theorem to Real-World Problems

Pythagoras' Theorem can be used to solve many practical, real-world problems that can be modelled as right-angled triangles. For example, to find the length of a ladder needed to reach a window 8 m above the ground, if the base of the ladder is placed 6 m away from the wall, the ladder itself forms the hypotenuse: length = √(8² + 6²) = √(64 + 36) = √100 = 10 m. Similar reasoning applies to finding diagonal distances across rectangular fields, calculating the shortest distance between two points, and determining the length of support beams or cables in construction.

Finding the Diagonal of a Rectangle

Since a diagonal line drawn across a rectangle divides it into two right-angled triangles, Pythagoras' Theorem can be used to find the length of the diagonal if the length and width of the rectangle are known. For a rectangle with length 12 cm and width 5 cm, the diagonal is calculated as d = √(12² + 5²) = √(144 + 25) = √169 = 13 cm.

Distance Between Two Points on a Grid

Pythagoras' Theorem is also the basis for calculating the straight-line distance between two points on a coordinate grid. If two points have a horizontal difference of Δx and a vertical difference of Δy, the distance between them is calculated as distance = √(Δx² + Δy²), which is directly derived from Pythagoras' Theorem, treating the horizontal and vertical differences as the two legs of a right-angled triangle, and the direct distance between the points as the hypotenuse.

Historical Background

Although named after the ancient Greek mathematician Pythagoras (who lived around the 6th century BC), evidence suggests that the relationship described by this theorem was known and used by earlier civilisations, including the Babylonians and Egyptians, particularly for practical purposes such as land surveying and construction. Pythagoras and his followers are credited with providing one of the earliest known formal mathematical proofs of the theorem.

Common Mistakes with Pythagoras' Theorem

Common errors when applying Pythagoras' Theorem include incorrectly identifying the hypotenuse (remember, it is always the side opposite the right angle, and always the longest side), forgetting to take the square root after adding or subtracting the squared values, applying the theorem to triangles that are not right-angled (the theorem only works for right-angled triangles), and making arithmetic errors when squaring numbers or simplifying square roots.

Summary

Pythagoras' Theorem describes the relationship between the three sides of a right-angled triangle: the square of the hypotenuse equals the sum of the squares of the other two sides (c² = a² + b²). This theorem allows us to find a missing side of a right-angled triangle when the other two sides are known, to verify whether a triangle is right-angled using its converse, and to solve a wide range of practical problems involving distances, heights, and diagonal measurements, making it one of the most useful and widely applied theorems in all of mathematics.

Practising Pythagoras' Theorem

Fluency with Pythagoras' Theorem comes from working through many varied problems, including finding a missing hypotenuse, finding a missing leg, verifying whether a triangle is right-angled, and applying the theorem to real-world word problems involving ladders, diagonals, and distances. Memorising a few common Pythagorean triples, such as (3, 4, 5) and (5, 12, 13), can save valuable time in examinations, but students should also practise problems with non-integer answers, where the final step requires simplifying or approximating a square root carefully.

Worked Examples

Example 1: Find the hypotenuse of a right triangle with legs 6 cm and 8 cm. c = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.

Example 2: Find the missing leg of a right triangle with hypotenuse 17 cm and one leg 8 cm. a = √(17² − 8²) = √(289 − 64) = √225 = 15 cm.

Example 3: Verify whether a triangle with sides 9 cm, 12 cm, and 15 cm is right-angled. 15² = 225, and 9² + 12² = 81 + 144 = 225. Since both are equal, the triangle is right-angled.

Example 4: A ladder reaches a window 15 m above the ground, with its foot placed 8 m from the wall. Find the length of the ladder. Length = √(15² + 8²) = √(225 + 64) = √289 = 17 m.

Example 5: Find the diagonal of a rectangle 15 cm by 8 cm. d = √(15² + 8²) = √(225 + 64) = √289 = 17 cm.

Student Exercise

Solve the following problems, showing all your working:

1. Find the hypotenuse of a right triangle with legs 9 cm and 12 cm.
2. Find the missing leg of a right triangle with hypotenuse 25 cm and one leg 7 cm.
3. Verify whether a triangle with sides 7 cm, 24 cm, and 25 cm is right-angled.
4. A ladder reaches a window 15 m high, with its foot 8 m from the wall. Find the length of the ladder.
5. Find the diagonal of a rectangle 20 cm by 21 cm.
6. Find the distance between the points (0, 0) and (6, 8) on a coordinate grid.
7. Find the missing leg of a right triangle with hypotenuse 13 cm and one leg 5 cm.
8. Find the hypotenuse of a right triangle with legs 5 cm and 12 cm.
9. Verify whether a triangle with sides 10 cm, 24 cm, and 26 cm is right-angled.
10. A rectangular field measures 30 m by 40 m. Find the length of a diagonal path across it.