Pythagoras Theorem Explained
Pythagoras theorem states that in a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides. It is one of the most im
TL;DR: Pythagoras theorem states that in a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides. It is one of the mo…
Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated
Pythagoras Theorem: Pythagoras theorem states that in a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides. It is one of the most im
Pythagoras Theorem Explained Simply
Pythagoras theorem states that in a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides. It is one of the most im This is a key Mathematics concept for Class 8-9.
Statement and Formula
Pythagoras theorem applies to right triangles which have one 90-degree angle. In a right triangle, the side opposite the right angle is called the hypotenuse, and the other two sides are called legs. The theorem states that the square of the hypotenuse equals the sum of the squares of the legs. If the legs are a and b and the hypotenuse is c, then the formula is a squared plus b squared equals c squared, or a^2 + b^2 = c^2.
Proof of Pythagoras Theorem
One common proof uses area. Consider a square with side length a plus b. Inside this large square, place a right triangle with legs a and b in each corner. The four triangles and one central square with side c must fill the large square. The area of the large square is (a + b)^2 which equals a^2 + 2ab + b^2. The area of four triangles is 4 times (half of a times b) which equals 2ab. The central square has area c^2. Since areas must be equal: a^2 + 2ab + b^2 equals 2ab plus c^2. Simplifying gives a^2 + b^2 equals c^2.
Pythagorean Triples
A Pythagorean triple is a set of three positive integers a, b, and c that satisfy the equation a^2 + b^2 = c^2. Common Pythagorean triples include 3, 4, 5 where 3^2 + 4^2 equals 9 + 16 equals 25 equals 5^2. Another is 5, 12, 13 where 5^2 + 12^2 equals 25 + 144 equals 169 equals 13^2. Multiples of these are also Pythagorean triples, so 6, 8, 10 and 9, 12, 15 are also valid. These triples are useful because they give exact integer solutions.
Finding Missing Sides
If you know two sides of a right triangle, you can find the third using the theorem. If you know both legs a and b, calculate c equals the square root of (a^2 + b^2). If you know the hypotenuse c and one leg a, find the other leg b equals the square root of (c^2 - a^2). For example, if a equals 3 and b equals 4, then c equals the square root of (9 + 16) equals the square root of 25 equals 5. If c equals 13 and a equals 5, then b equals the square root of (169 - 25) equals the square root of 144 equals 12.
Distance Formula and Applications
The Pythagoras theorem is the basis for the distance formula between two points. If points are at coordinates (x1, y1) and (x2, y2), the distance d equals the square root of ((x2 - x1)^2 + (y2 - y1)^2). This formula comes directly from Pythagoras theorem where the differences in x and y coordinates are the legs and the distance is the hypotenuse. This is used extensively in coordinate geometry, physics, surveying, construction, and navigation.
Worked Examples
Example 1. A ladder 13 meters long leans against a wall. The base of the ladder is 5 meters from the wall. How high up the wall does the ladder reach?
Solution: Let a equal distance from wall = 5 meters, c equal ladder length = 13 meters, and b equal height on wall = unknown. Using a^2 + b^2 = c^2: 5^2 + b^2 = 13^2. 25 + b^2 = 169. b^2 = 144. b = 12 meters. The ladder reaches 12 meters up the wall.
Example 2. Check if a triangle with sides 7, 8, and 9 is a right triangle.
Solution: To be a right triangle, the largest side 9 must be the hypotenuse. Check: 7^2 + 8^2 equals 49 + 64 equals 113, which is not equal to 9^2 which equals 81. Since 113 is not equal to 81, this is not a right triangle.
Example 3. Find the distance between points A(1, 2) and B(4, 6).
Solution: Using the distance formula d = square root of ((x2 - x1)^2 + (y2 - y1)^2). d = square root of ((4 - 1)^2 + (6 - 2)^2). d = square root of (3^2 + 4^2). d = square root of (9 + 16). d = square root of 25 = 5 units.
Where You See This in Real Life
- Construction workers use Pythagoras theorem to make sure buildings have perfect right angles. A 3-4-5 triangle guarantees a 90-degree angle.
- Surveyors use the theorem to measure distances across rivers or mountains that cannot be directly measured. They measure two perpendicular distances and calculate the diagonal.
- Architects use the theorem to design roof trusses and calculate the length of beams needed for structural support.
- GPS and mapping use distance formulas based on Pythagoras theorem to calculate distances between locations on Earth.
Common Mistakes Students Make
- Confusing which side is the hypotenuse - it is always the longest side opposite the right angle
- Forgetting to take the square root when solving for c - after getting c^2 = value, you must take the square root
- Using the theorem in triangles without a right angle - the theorem only applies to right triangles
- Making arithmetic errors when squaring numbers - double-check calculations like 7^2 = 49, not 47
Frequently Asked Questions
Who was Pythagoras and why is the theorem named after him?
Pythagoras was an ancient Greek mathematician and philosopher who lived around 570 to 495 BCE. Although the relationship was known before him, he is credited with proving the theorem and understanding its significance. The theorem is named in his honor.
Does the theorem work for all triangles?
No, Pythagoras theorem only works for right triangles that have exactly one 90-degree angle. For other triangles, there are different relationships called the Law of Cosines.
Why is 3-4-5 the most famous Pythagorean triple?
The 3-4-5 triple is the smallest Pythagorean triple with the simplest numbers. It is easy to remember and use. Construction workers especially use it because it is easy to measure in the field.