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Mathematics · Class 10 · Chapter 23: Pythagoras' Theorem

8 min

Key points

Pythagoras' theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides
The theorem is written as: a² + b² = c², where c is the hypotenuse and a, b are the perpendicular sides
Converse of Pythagoras' theorem: If the square of one side of a triangle equals the sum of squares of the other two sides, then the triangle is right-angled
Pythagorean triplets are sets of three positive integers that satisfy the Pythagorean theorem (e.g., 3-4-5, 5-12-13)
The theorem is used to find the third side when two sides of a right triangle are known
In practical applications, it helps calculate distances, heights, and lengths in construction and surveying
The altitude drawn to the hypotenuse creates two smaller triangles similar to the original and to each other
Common Pythagorean triplets include: (3,4,5), (5,12,13), (8,15,17), (7,24,25), (9,40,41)

Definitions

Pythagoras' Theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Hypotenuse: The longest side of a right-angled triangle, opposite to the right angle.
Pythagorean Triplet: A set of three positive integers a, b, c such that a² + b² = c².
Converse of Pythagoras' Theorem: If in a triangle, the square of one side equals the sum of squares of the other two sides, then the triangle is right-angled.

Worked example

State Pythagoras' theorem and find the length of hypotenuse of a right triangle whose other two sides are 5 cm and 12 cm. (5 marks)

Pythagoras' theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides. Using c² = a² + b², where c is hypotenuse, we get c² = 5² + 12² = 169, so c = 13 cm.

Common mistakes

Students often forget to take square root at the end when finding a side length
Confusing which side is the hypotenuse - always remember it's the longest side opposite to right angle
Not writing the formula before substituting values in numerical problems
Making calculation errors while squaring and finding square roots
Using Pythagoras' theorem for non-right triangles without checking

Quick recap

Pythagoras' theorem applies only to right triangles. The formula is c² = a² + b² where c is the hypotenuse (longest side opposite to right angle). The converse is also true - if this relationship holds, the triangle is right-angled. Always state the theorem exactly as in textbook for definition marks.

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