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State Board · Class 6 · Mathematics

Lesson 16: Baudhayana-Pythagoras Theorem

5 concepts · 40 practice questions with answers · Free to read

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Right-angled triangle and its sides

From daily life

A ladder leaning on a wall, the corner of a book and a set square all show a right angle. The triangle formed has a special name and a special rule.

Simple definition

A triangle with one angle of 90° is a right-angled triangle (काटकोन त्रिकोण). The side opposite the right angle is the hypotenuse (कर्ण).

Rule 1: Sides of a right-angled triangle — In ∆ABC with ∠B = 90°, the hypotenuse is AC. The two sides that form the right angle are AB and BC.

Right-angled triangle PQR: right angle at Q, hypotenuse PR.PQRPQQRPR = hypotenuse90°Right angle at QSides at Q:PQ, QRHypotenuse isthe longest side

In the figure : In the figure the right-angled triangle and the hypotenuse.

  • Easy : Easy: In ∆DEF with ∠E = 90°, the hypotenuse is DF.
  • Medium : Medium: In ∆XYZ with ∠Y = 90°, the sides XY and YZ form the right angle.
  • Exam-type : Exam: The set square in your box has a 90° corner, so it is a right-angled triangle.

Rule 2: The hypotenuse and the other two angles — The hypotenuse is always the longest side. A triangle can have only one right angle, so the other two angles add up to 90°. If one is 35°, the other is 90° − 35° = 55°.

  • Easy : Easy: Hypotenuse is the longest side.
  • Medium : Medium: Angles 90° and 48° leave 180° − 138° = 42°.
  • Exam-type : Exam: Two right angles in one triangle would already be 180°, leaving no room for the third angle; so a triangle has only one.
Remember

Hypotenuse is opposite 90° and is the longest. The other two angles add up to 90°.

Common mistake

Calling a leg the hypotenuse. The hypotenuse is opposite the right angle.

Shortcut

Memory clue: "काटकोनासमोरची बाजू = कर्ण".

Spot the difference

PartIn ∆ABC (∠B = 90°)
HypotenuseAC
Sides forming right angleAB, BC

Practice questions

Easy
  1. 1Which side is the hypotenuse?
  2. 2In ∆DEF with ∠E = 90°, name the hypotenuse.
  3. 3Name the sides forming the right angle in ∆DEF.
Medium
  1. 4What do the two non-right angles of a right triangle add up to?
  2. 5One acute angle of a right triangle is 35°. Find the other.
  3. 6Can a triangle have two right angles?
Challenging
  1. 7Which is the longest side of a right-angled triangle?
  2. 8A triangle has angles 90° and 48°. Find the third angle.
Show answers (try them yourself first)
  1. 1The side opposite the right angle.
  2. 2DF.
  3. 3DE and EF.
  4. 490°.
  5. 555°.
  6. 6No, they would add up to 180° already.
  7. 7The hypotenuse.
  8. 842°.

The Baudhayana-Pythagoras theorem

From daily life

A carpenter wants to check if a corner of a cupboard is exactly square. He measures along the two edges and then across the corner. A rule of lengths tells him whether the corner is a right angle.

Simple definition

Baudhayana-Pythagoras theorem (बौधायन-पायथागोरस प्रमेय): in a right-angled triangle, (hypotenuse)² = (side 1)² + (side 2)².

Rule 1: The theorem with squares — The squares on the two shorter sides together have the same area as the square on the hypotenuse. With sides 6 and 8: 36 + 64 = 100, which is 10². So the hypotenuse is 10.

Squares on the sides of a 3-4-5 right triangle have areas 9, 16 and 25.16925433² + 4² = 5²9 + 16 = 25(hypotenuse)² =sum of squaresof the other two

In the figure : In the figure the squares on the sides: 9 + 16 = 25.

Live figure

The Baudhayana-Pythagoras theorem

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  • Easy : Easy: 6² + 8² = 36 + 64 = 100 = 10².
  • Medium : Medium: 9² + 12² = 81 + 144 = 225 = 15².
  • Exam-type : Exam: Is a triangle with sides 7, 9, 12 right-angled? 49 + 81 = 130, but 12² = 144. No.

Rule 2: History and limits — Baudhayana stated this rule in the Shulvasutra around the 8th century BC. The Greek Pythagoras (570–495 BC) formulated and proved it. It works only for right-angled triangles.

  • Easy : Easy: The theorem works for right-angled triangles only.
  • Medium : Medium: Baudhayana lived much before Pythagoras.
  • Exam-type : Exam: The squares on legs 7 and 24 have areas 49 and 576, which add to 625 = 25², so the hypotenuse is 25.
Remember

Hypotenuse² = side₁² + side₂². Only for right-angled triangles.

Common mistake

Using the theorem for any triangle. It needs a right angle.

Shortcut

Memory clue: "दोन बाजूंचे वर्ग मिळून कर्णाचा वर्ग".

Spot the difference

ItemDetail
Rulec² = a² + b²
Stated byBaudhayana, 8th century BC
Proved byPythagoras, 570–495 BC

Practice questions

Easy
  1. 1State the theorem in words.
  2. 2Find 6² + 8².
  3. 3Legs are 9 and 12. Find the square of the hypotenuse.
Medium
  1. 4Who stated this rule around the 8th century BC?
  2. 5Who proved it?
  3. 6Does the theorem work for all triangles?
Challenging
  1. 7Squares on the legs have areas 36 and 64. What is the area of the square on the hypotenuse?
  2. 8Squares on the legs have areas 49 and 576. Find the hypotenuse.
Show answers (try them yourself first)
  1. 1In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
  2. 2100.
  3. 381 + 144 = 225.
  4. 4Baudhayana.
  5. 5Pythagoras.
  6. 6No, only for right-angled triangles.
  7. 7100.
  8. 8625 = 25², so 25.

Finding the hypotenuse

From daily life

A boy walks 6 m west and then 8 m south. How far is he from the start in a straight line? The path makes a right angle, so the theorem finds the distance.

Simple definition

To find the hypotenuse, add the squares of the two sides forming the right angle, then take the square root.

Rule 1: Finding the hypotenuse — Sides 8 and 15: 64 + 225 = 289 = 17², so the hypotenuse is 17. Sides 6 and 8: 36 + 64 = 100 = 10², so the hypotenuse is 10.

Finding the hypotenuse for sides 5 and 12.512?Find the hypotenuse5² + 12²= 25 + 144= 169 = 13²Hypotenuse = 13Other side: subtract

In the figure : In the figure finding the hypotenuse.

  • Easy : Easy: Sides 6 and 8 give hypotenuse 10.
  • Medium : Medium: Sides 8 and 15 give hypotenuse 17.
  • Exam-type : Exam: Sides 20 and 21: 400 + 441 = 841 = 29², so the hypotenuse is 29.

Rule 2: Where it is used — A boy walking 6 m west and 8 m south is 10 m from the start in a straight line. The diagonal of a rectangle is the hypotenuse of a right triangle: a rectangle 8 cm by 15 cm has diagonal 17 cm.

  • Easy : Easy: Walk 6 m west then 8 m south: distance 10 m.
  • Medium : Medium: Rectangle 8 cm by 15 cm: diagonal 17 cm.
  • Exam-type : Exam: Legs 10 and 24: 100 + 576 = 676 = 26², so the hypotenuse is 26.
Remember

Add the squares, then take the square root.

Common mistake

Forgetting the square root and giving 289 as the hypotenuse. It is 17.

Shortcut

Memory clue: "वर्ग जोडा, मग मूळ काढा".

Spot the difference

LegsHypotenuse
6, 810
8, 1517
20, 2129

Practice questions

Easy
  1. 1Legs are 6 and 8. Find the hypotenuse.
  2. 2Legs are 8 and 15. Find the hypotenuse.
  3. 3Legs are 20 and 21. Find the hypotenuse.
Medium
  1. 4Legs are 10 and 24. Find the hypotenuse.
  2. 5A girl walks 15 m north and then 8 m east. How far is she from the start?
  3. 6A rectangle is 15 cm by 20 cm. Find its diagonal.
Challenging
  1. 7Legs are 12 and 16. Find the hypotenuse.
  2. 8Why do we take the square root at the end?
Show answers (try them yourself first)
  1. 110.
  2. 217.
  3. 329.
  4. 426.
  5. 5225 + 64 = 289, so 17 m.
  6. 6225 + 400 = 625, so 25 cm.
  7. 7144 + 256 = 400, so 20.
  8. 8The sum of the squares gives (hypotenuse)², not the hypotenuse.

Finding a missing side

From daily life

A 26 m ladder leans on a wall and its top touches a window 24 m above the ground. How far is the foot of the ladder from the wall? The hypotenuse is known, so we subtract.

Simple definition

When the hypotenuse is known: (side)² = (hypotenuse)² − (other side)².

Rule 1: Finding a missing side — Hypotenuse 17 and a side 8: 289 − 64 = 225 = 15², so the other side is 15. Hypotenuse 29 and a side 21: 841 − 441 = 400 = 20², so the other side is 20.

  • Easy : Easy: Hypotenuse 10 and a side 6: 100 − 36 = 64, so the other side is 8.
  • Medium : Medium: Hypotenuse 17 and a side 8 give 15.
  • Exam-type : Exam: Hypotenuse 29, a side 21 give the other side 20.

Rule 2: The ladder problem — A 26 m ladder reaches a window 24 m high. The foot is at distance d, where d² = 26² − 24² = 676 − 576 = 100, so d = 10 m.

A 35 ft ladder reaching 21 ft up a wall has its foot 28 ft away.35 ft21 ft? = 28 ftwall35² − 21²= 1225 − 441= 784 = 28²

In the figure : In the figure the ladder problem.

  • Easy : Easy: A 10 m ladder, wall height 8 m: foot distance = 6 m (100 − 64 = 36).
  • Medium : Medium: 26 m ladder, 24 m high: foot at 10 m.
  • Exam-type : Exam: A rectangle has diagonal 13 cm and one side 5 cm; the other side is √(169 − 25) = √144 = 12 cm.
Remember

Hypotenuse known: subtract. Check: the hypotenuse is longer than each other side.

Common mistake

Adding the squares when the hypotenuse is given. Subtract.

Shortcut

Memory clue: "कर्ण माहीत — वजा करा; कर्ण शोधायचा — बेरीज करा".

Spot the difference

KnownMethod
Both legsAdd squares
Hypotenuse and one legSubtract squares

Practice questions

Easy
  1. 1Hypotenuse 5 and a side 3. Find the other side.
  2. 2Hypotenuse 13 and a side 12. Find the other side.
  3. 3Hypotenuse 17 and a side 15. Find the other side.
Medium
  1. 4Hypotenuse 25 and a side 24. Find the other side.
  2. 5Hypotenuse 20 and a side 16. Find the other side.
  3. 6A 10 m ladder touches a wall at a height of 8 m. How far is the foot from the wall?
Challenging
  1. 7A rectangle has diagonal 13 cm and one side 5 cm. Find the other side.
  2. 8Why should the hypotenuse be longer than the other sides in the answer?
Show answers (try them yourself first)
  1. 125 − 9 = 16, so 4.
  2. 2169 − 144 = 25, so 5.
  3. 3289 − 225 = 64, so 8.
  4. 4625 − 576 = 49, so 7.
  5. 5400 − 256 = 144, so 12.
  6. 6100 − 64 = 36, so 6 m.
  7. 7169 − 25 = 144, so 12 cm.
  8. 8It is the longest side, so if not, there is an error.

Triplets and right-angled triangles

From daily life

Three numbers such as 6, 8 and 10 fit the rule exactly. Such sets of numbers are called Pythagorean triplets, and builders use them to make perfect right angles.

Simple definition

Three natural numbers where the square of the largest equals the sum of the squares of the other two form a triplet (त्रिकूट).

Rule 1: Triplets — (6, 8, 10), (8, 15, 17) and (20, 21, 29) are triplets because 6² + 8² = 10², 8² + 15² = 17² and 20² + 21² = 29². A triangle with these sides is right-angled.

Checking Baudhayana-Pythagorean triplets.NumbersCheck3, 4, 59 + 16 = 25✔7, 24, 2549 + 576 = 625✔15, 20, 25225 + 400 = 625✔11, 12, 15121 + 144 = 265 ≠ 225✘Doubling (5, 12, 13) → (10, 24, 26) is also a triplet

In the figure : In the figure checking triplets.

  • Easy : Easy: (6, 8, 10): 36 + 64 = 100 = 10² ✓.
  • Medium : Medium: (8, 15, 17): 64 + 225 = 289 = 17² ✓.
  • Exam-type : Exam: (9, 12, 16): 81 + 144 = 225, but 16² = 256. Not a triplet.

Rule 2: Making new triplets — If every number of a triplet is multiplied by the same number, we again get a triplet. (3, 4, 5) × 3 = (9, 12, 15) and (8, 15, 17) × 2 = (16, 30, 34).

  • Easy : Easy: (3, 4, 5) × 2 = (6, 8, 10).
  • Medium : Medium: (3, 4, 5) × 4 = (12, 16, 20); 144 + 256 = 400 ✓.
  • Exam-type : Exam: (16, 30, 34): 256 + 900 = 1156 = 34² ✓.
Remember

Test: smaller squares add to the largest square. Multiples of a triplet are triplets.

Common mistake

Testing with the wrong number as the largest. The largest number always goes alone on the right.

Shortcut

Memory clue: "लहान दोन वर्ग = मोठा वर्ग".

Spot the difference

TripletCheck
(6, 8, 10)36 + 64 = 100
(8, 15, 17)64 + 225 = 289
(9, 12, 16)81 + 144 ≠ 256

Practice questions

Easy
  1. 1Is (6, 8, 10) a triplet?
  2. 2Is (5, 12, 13) a triplet?
  3. 3Is (7, 8, 9) a triplet?
Medium
  1. 4Multiply (3, 4, 5) by 4. What do you get?
  2. 5Is (20, 21, 29) a triplet?
  3. 6Find the missing number in (16, 30, ?).
Challenging
  1. 7Is a triangle with sides 10, 24, 26 right-angled?
  2. 8Is a triangle with sides 7, 9, 11 right-angled? Why?
Show answers (try them yourself first)
  1. 1Yes, 36 + 64 = 100.
  2. 2Yes, 25 + 144 = 169.
  3. 3No, 49 + 64 = 113 but 9² = 81.
  4. 4(12, 16, 20).
  5. 5Yes, 400 + 441 = 841 = 29².
  6. 6256 + 900 = 1156 = 34², so 34.
  7. 7Yes, 100 + 576 = 676 = 26².
  8. 8No, 49 + 81 = 130 but 11² = 121.