What this chapter is about
This chapter introduces one of the most important results in all of mathematics: the relationship between the sides of a right-angled triangle. Ancient Indian mathematician Baudhayana described this relationship in his Sulbasutras around 800 BCE, and Greek mathematician Pythagoras studied it later around 500 BCE. Both contributions are honoured in the chapter title.
The theorem states that in any right-angled triangle, the square of the longest side (called the hypotenuse) equals the sum of the squares of the other two sides. This simple relationship has enormous practical value — builders use it to check if corners are perfectly square, surveyors use it to measure distances, and it forms the foundation for coordinate geometry you will study later.
After studying this chapter, you should be able to identify right-angled triangles, apply the theorem to find unknown sides, verify whether a triangle is right-angled using the converse of the theorem, and solve real-world problems involving right triangles.
Key ideas
- In a right-angled triangle, the side opposite the right angle is called the hypotenuse. It is always the longest side.
- The Baudhayana-Pythagoras Theorem states: In a right-angled triangle, (hypotenuse)² = (base)² + (height)², where base and height are the two sides forming the right angle.
- The converse is also true: If the square of one side of a triangle equals the sum of squares of the other two sides, the triangle must be right-angled.
- A Pythagorean triplet is a set of three positive whole numbers (a, b, c) where a² + b² = c². Common examples are (3, 4, 5), (5, 12, 13), (8, 15, 17) and (7, 24, 25).
- If you multiply each number in a Pythagorean triplet by the same whole number, you get another Pythagorean triplet. For example, multiplying (3, 4, 5) by 2 gives (6, 8, 10).
- The theorem works only for right-angled triangles. For other triangles, the relationship does not hold exactly.
Formulas and facts to remember
The Baudhayana-Pythagoras Theorem: c² = a² + b² Here c is the hypotenuse, and a and b are the other two sides of a right-angled triangle.
Finding the hypotenuse: c = √(a² + b²) Use this when you know both shorter sides and need the longest side.
Finding a shorter side: a = √(c² − b²) Use this when you know the hypotenuse and one shorter side.
Test for a right-angled triangle: If the three sides satisfy (longest side)² = (side 1)² + (side 2)², the triangle is right-angled.
Generating Pythagorean triplets: For any whole number m greater than 1: the numbers (m² − 1), (2m), and (m² + 1) form a Pythagorean triplet.
Worked examples
Example 1: Finding the hypotenuse
A ladder is placed against a wall. The foot of the ladder is 6 metres from the wall, and the ladder reaches 8 metres up the wall. How long is the ladder?
The wall, ground, and ladder form a right-angled triangle. The ladder is the hypotenuse.
Using the theorem: Ladder² = 6² + 8² Ladder² = 36 + 64 Ladder² = 100 Ladder = √100 = 10 metres
The ladder is 10 metres long.
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Example 2: Finding a shorter side
A rectangular cricket pitch has a diagonal of 25 metres. If the length of the pitch is 24 metres, find its width.
The diagonal, length, and width form a right-angled triangle. The diagonal is the hypotenuse.
Using the theorem: 25² = 24² + width² 625 = 576 + width² width² = 625 − 576 = 49 width = √49 = 7 metres
The width of the pitch is 7 metres.
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Example 3: Checking if a triangle is right-angled
Ravi cuts three sticks of lengths 9 cm, 40 cm, and 41 cm. Can he form a right-angled triangle with these sticks?
Check if (longest side)² = sum of squares of other two sides.
41² = 1681 9² + 40² = 81 + 1600 = 1681
Since 1681 = 1681, the three lengths form a right-angled triangle.
Common mistakes
Using the theorem for triangles that are not right-angled → First confirm there is a 90° angle before applying the formula.
Adding all three squares instead of two → Only the two shorter sides are squared and added; the hypotenuse is squared alone on one side of the equation.
Forgetting to take the square root at the end → After finding c² = 100, students sometimes write the answer as 100 instead of √100 = 10.
Identifying the wrong side as hypotenuse → The hypotenuse is always opposite the right angle and is always the longest; check before substituting values.
Assuming any three numbers form a Pythagorean triplet → Verify by calculation; for example, 3, 5, 7 do not satisfy 3² + 5² = 7² since 9 + 25 = 34 ≠ 49.
Quick revision
- In a right triangle: (hypotenuse)² = (base)² + (height)².
- The hypotenuse is opposite the 90° angle and is always the longest side.
- To test if a triangle is right-angled, check if the square of the longest side equals the sum of squares of the other two.
- (3, 4, 5), (5, 12, 13), and (8, 15, 17) are common Pythagorean triplets — memorise them for quick checks.
- Always take the square root after finding the square of the unknown side.