Chapter focus
The Baudhayana–Pythagoras theorem connects the side lengths of a right-angled triangle. A diagram and the right-angle condition are essential when applying it.
These original notes and problems support revision. Follow your assigned Ganita Prakash Part II for its full geometric investigations and exercises.
1. The theorem
In any right-angled triangle, let a and b be the legs that meet at the right angle, and let c be the hypotenuse opposite the right angle. Then:
a2 + b2 = c2.
The hypotenuse is always the longest side of a right triangle. The theorem describes a relation among the areas of squares drawn on the three sides: the areas on the legs together equal the area on the hypotenuse.
This relationship applies only to right-angled triangles. Do not use it just because a triangle looks right-angled in a sketch.
2. Find the hypotenuse
Worked example: legs 6 cm and 8 cm
- Identify the two perpendicular legs: a = 6 and b = 8.
- Substitute: c2 = 62 + 82 = 36 + 64 = 100.
- Take the positive square root for a length: c = √100 = 10 cm.
- Check: 6, 8, 10 satisfy 36 + 64 = 100, and 10 is the longest side.
3. Find a missing leg
If the hypotenuse and one leg are known, rearrange the equation: a2 = c2 − b2. The hypotenuse must be the longest side, so the subtraction should produce a positive value.
Worked example: hypotenuse 13 m, one leg 5 m
Let the missing leg be x. Then x2 + 52 = 132. Thus x2 = 169 − 25 = 144, so x = 12 m. Check: 52 + 122 = 25 + 144 = 169.
4. Use the converse to test a triangle
If the squares of the two shorter sides add to the square of the longest side, the triangle is right-angled. This is the converse of the theorem.
Example: For side lengths 9, 12 and 15, 92 + 122 = 81 + 144 = 225 = 152. Therefore the triangle is right-angled, with hypotenuse 15.
Always test the two shorter sides against the longest side. A random choice of sides can lead to a false conclusion.
5. Apply the theorem to distance problems
Draw a right triangle that matches the situation and label the perpendicular directions. A rectangle’s diagonal, a ladder against a wall, and the shortest straight route across a rectangular field can form right triangles.
Worked example: a ladder
A 13 m ladder reaches a point 12 m above level ground. The wall and ground meet at a right angle. If the foot is x metres from the wall, then x2 + 122 = 132; x2 = 25; x = 5 m. The answer is a length, so it must be positive.
Keep units consistent before squaring. If one side is given in centimetres and another in metres, convert them to the same unit first.
Common errors to avoid
- The hypotenuse is opposite the right angle, not simply the side drawn at the bottom.
- Use c2 as the largest squared value; do not add the hypotenuse to a leg.
- When finding a leg, subtract squares first, then take the square root.
- For a length, take the positive root and include the correct unit.
Practice: draw and label a triangle
1. Find the hypotenuse when the legs are 5 cm and 12 cm.
Answer: c2 = 25 + 144 = 169, so c = 13 cm.
2. A 13 m ladder reaches 12 m up a wall. How far is its foot from the wall?
Answer: x2 = 132 − 122 = 25, so x = 5 m.
3. Is a triangle with sides 7, 24 and 25 right-angled?
Answer: Yes. 72 + 242 = 49 + 576 = 625 = 252.
4. A rectangle measures 9 cm by 12 cm. Find its diagonal.
Answer: d2 = 92 + 122 = 225, so d = 15 cm.
5. A right triangle has hypotenuse 10 cm and one leg 8 cm. Find the other leg.
Answer: x2 = 100 − 64 = 36, so x = 6 cm.
Chapter summary
In a right triangle, the squares on the perpendicular legs sum to the square on the hypotenuse. Use the theorem to find a missing side, and its converse to test whether a triangle is right-angled. Draw the geometry, label the hypotenuse and check that the result is reasonable.