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CBSE · Class 8 · Mathematics Part 2

Part 2, Chapter 2: The Baudhāyana-Pythagoras Theorem

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

This lesson is also in: मराठी · हिंदी

Doubling and halving a square

From daily life

A tile-maker has a square tile and wants a new square tile with exactly double the area. He does not measure anything: he draws a square on the diagonal of the old tile.

Simple definition

The square drawn on the diagonal of a square has DOUBLE the area. Joining the midpoints of the sides of a square gives a square with HALF the area. This idea is the seed of the Baudhāyana-Pythagoras theorem.

5025cm²

In the figure: Joining the midpoints cuts the area 50 to 25. Going the other way, the big square is the square on the diagonal of the tilted one.

Rule 1: Square on the diagonal — Its area is twice the area of the original square.

  • Easy: Side 5 cm: the diagonal square has area 2 × 25 = 50 cm².
  • Medium: Area 18 cm²: the diagonal square has area 36 cm².
  • Exam-type: Side 12 cm: area 144 becomes 288 cm², and the new side is 12√2 cm.

Rule 2: Join the midpoints — The inner tilted square has exactly half the area.

  • Easy: Area 72 cm² gives 36 cm².
  • Medium: Area 200 cm², joined twice: 100, then 50 cm².
  • Exam-type: Side 10 cm: the inner square has area 50, so its side is 5√2 cm.

Rule 3: Double side, double area? — No. Doubling the side makes the area 4 times. To get double the area, multiply the side by √2.

  • Easy: Side 3: area 9; side 6: area 36 (4 times).
  • Medium: For double the area of a 3 cm square use side 3√2 cm ≈ 4.2 cm.
  • Exam-type: Halve the area of a square: multiply the side by 1/√2.
Remember

Cut a square along both diagonals to get 4 equal triangles. The original square holds 4 of them and the square on its diagonal holds 8, so it is twice as large.

Common mistake

Thinking that doubling the side doubles the area. Doubling the side makes the area 4 times.

Shortcut

Diagonal doubles, midpoints halve. Memory clue: "Diagonal = Double, Midpoints = Half".

Spot the difference

PointSquare on the diagonalSquare from midpoints
Area2 × original½ × original
Example (original 50)10025
Side compared with the originalside × √2 (longer)side ÷ √2 (shorter)

Practice questions

Easy
  1. 1A square has side 8 cm. Find the area of the square on its diagonal.
  2. 2The midpoints of a square of area 60 cm² are joined. Find the area of the new square.
  3. 3Find the side of a square with double the area of a square of side 6 cm.
Medium
  1. 4A square has area 98 cm². Find the area of the square on its diagonal.
  2. 5The side of a square is doubled. How many times does its area become?
  3. 6Midpoints of a square of area 80 cm² are joined, then midpoints of the new square are joined. Find the last area.
Challenging
  1. 7A square has side 10 cm. Midpoints of its sides are joined. Find the side of the inner square.
  2. 8The square on the diagonal of a square has area 128 cm². Find the side of the original square.
Show answers (try them yourself first)
  1. 1128 cm²
  2. 230 cm²
  3. 36√2 cm
  4. 4196 cm²
  5. 54 times
  6. 620 cm²
  7. 75√2 cm
  8. 88 cm

Isosceles right triangle and the diagonal of a square

From daily life

A square garden of side 40 m has a path across the corner. Walking along the diagonal is shorter than walking along two sides, but how much longer than one side is it? The answer is √2 times.

Simple definition

In an isosceles right triangle the two equal sides (legs) meet at the right angle and the other two angles are 45° each. The hypotenuse = leg × √2. A square cut along a diagonal gives two such triangles, so diagonal = side × √2.

555√245°45°

In the figure: Legs 5 and 5, so the hypotenuse is 5√2 ≈ 7.05.

Rule 1: Find the hypotenuse — Multiply the leg by √2 (use 1.41 for √2 when an approximate value is wanted).

  • Easy: Leg 4 cm: hypotenuse 4√2 ≈ 5.64 cm.
  • Medium: Square field of side 60 m: diagonal 60√2 ≈ 84.6 m.
  • Exam-type: Legs 9 cm: perimeter 18 + 9√2 ≈ 30.7 cm.

Rule 2: Find the leg — Divide the hypotenuse by √2.

  • Easy: Hypotenuse 10√2 cm: leg = 10 cm.
  • Medium: Hypotenuse 12 cm: leg = 12 ÷ √2 = 6√2 ≈ 8.46 cm.
  • Exam-type: Diagonal of a square is 20 cm: side = 10√2 ≈ 14.1 cm.

Rule 3: Area using the diagonal — Square on the diagonal has double the area, so area of the square = (diagonal)² ÷ 2.

  • Easy: Diagonal 10 cm: area = 100 ÷ 2 = 50 cm².
  • Medium: Diagonal 12 cm: area = 144 ÷ 2 = 72 cm².
  • Exam-type: Isosceles right triangle with hypotenuse 10 cm: legs 5√2, area = ½ × 50 = 25 cm².
Remember

√2 ≈ 1.41 is enough for most problems. The angles of this triangle are always 45°, 45° and 90°.

Common mistake

Multiplying the leg by 2 instead of √2. The hypotenuse is longer than a leg but shorter than the sum of both legs.

Shortcut

Leg → hypotenuse: multiply by √2; hypotenuse → leg: divide by √2. Memory clue: "One, one, root two".

Spot the difference

PointLeg to hypotenuseHypotenuse to leg
Operation× √2÷ √2
Example5 → 5√210√2 → 10
Result islongershorter

Practice questions

Easy
  1. 1The legs of an isosceles right triangle are 6 cm. Find the hypotenuse.
  2. 2The hypotenuse is 7√2 cm. Find each leg.
  3. 3Find the diagonal of a square of side 9 cm.
Medium
  1. 4Take √2 ≈ 1.41. Find the hypotenuse for legs of 20 cm.
  2. 5A square field has side 50 m. Find the diagonal (√2 ≈ 1.41).
  3. 6Legs are 10 cm. Find the perimeter.
Challenging
  1. 7An isosceles right triangle has hypotenuse 10 cm. Find its area.
  2. 8A square has diagonal 12 cm. Find its area.
Show answers (try them yourself first)
  1. 16√2 cm
  2. 27 cm
  3. 39√2 cm
  4. 428.2 cm
  5. 570.5 m
  6. 620 + 10√2 cm
  7. 725 cm²
  8. 872 cm²

The theorem and its converse

From daily life

A carpenter checks a door frame is square by measuring 9 cm along one edge, 12 cm along the other and the slanting distance between the marks. If it is 15 cm, the corner is a perfect right angle.

Simple definition

In a right triangle, the square on the hypotenuse equals the sum of the squares on the other two sides: a² + b² = c², where c is the hypotenuse. The converse is also true: if a² + b² = c² for three sides, the triangle is right-angled.

912159² + 12² = 15²

In the figure: 81 + 144 = 225 = 15², so the hypotenuse is 15.

Rule 1: Find the hypotenuse — Add the squares of the two shorter sides, then take the square root.

Live figure

The theorem and its converse

In the app you can move this figure with your finger.

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In the figure: Move the sliders: the area of the big square on the hypotenuse always equals the two small squares added together.

  • Easy: Legs 9 and 12: 81 + 144 = 225, so c = 15.
  • Medium: Legs 8 and 15: 64 + 225 = 289, so c = 17.
  • Exam-type: Legs 2 and 5: 4 + 25 = 29, so c = √29 ≈ 5.39.

Rule 2: Find a leg — Subtract the square of the known side from the square of the hypotenuse, then take the square root.

  • Easy: Hypotenuse 13, leg 5: 169 − 25 = 144, other leg 12.
  • Medium: Hypotenuse 26, leg 24: 676 − 576 = 100, other leg 10.
  • Exam-type: Hypotenuse 41, leg 9: 1681 − 81 = 1600, other leg 40.

Rule 3: Test with the converse — Square the longest side. Compare it with the sum of the squares of the other two.

  • Easy: 20, 21, 29: 400 + 441 = 841 = 29². Right-angled.
  • Medium: 10, 11, 15: 100 + 121 = 221, but 15² = 225. Not right-angled.
  • Exam-type: 9, 12, 16: 81 + 144 = 225, but 16² = 256. Not right-angled.
Remember

The hypotenuse is always the longest side and is opposite the right angle. In the converse test, always square the longest side.

Common mistake

Writing c² = a² − b² or adding the sides instead of their squares. Always use a² + b² = c² with c as the longest side.

Shortcut

Squares of the two small add up to the square of the big. Memory clue: "Little squares + little square = Big square".

Spot the difference

PointTheoremConverse
Starts witha right trianglethree side lengths
Concludesa² + b² = c²the triangle is right-angled
Used tofind a missing sidecheck a right angle

Practice questions

Easy
  1. 1The legs of a right triangle are 6 cm and 8 cm. Find the hypotenuse.
  2. 2The hypotenuse is 13 cm and one leg is 12 cm. Find the other leg.
  3. 3The legs are 15 cm and 20 cm. Find the hypotenuse.
Medium
  1. 4The hypotenuse is 26 cm and one leg 24 cm. Find the other leg.
  2. 5Is a triangle with sides 12, 35, 37 right-angled?
  3. 6The legs are 2 cm and 5 cm. Find the hypotenuse.
Challenging
  1. 7Is a triangle with sides 9, 12, 16 right-angled?
  2. 8The legs of a right triangle are 7 cm and 24 cm. Find its perimeter.
Show answers (try them yourself first)
  1. 110 cm
  2. 25 cm
  3. 325 cm
  4. 410 cm
  5. 5Yes (1225 + 144 = 1369 = 37²)
  6. 6√29 cm
  7. 7No (225 is not 256)
  8. 856 cm

Baudhāyana triples

From daily life

A rope-stretcher in a building site ties knots to make equal gaps and makes a loop of 12 gaps: 3 along one side, 4 along another and 5 across. The corner pulled tight is a right angle. Such sets of whole numbers are called triples.

Simple definition

A Baudhāyana (Pythagorean) triple is a set of three whole numbers a, b, c with a² + b² = c². Multiplying every number of a triple by the same number gives another triple. Famous ones: 3-4-5, 5-12-13, 8-15-17, 7-24-25.

3456810× 291215× 3152025× 5

In the figure: Every row is the first row multiplied by the same number, so every row is a triple.

Rule 1: Scale a triple — Multiply all three numbers by the same whole number.

  • Easy: 5, 12, 13 × 3 gives 15, 36, 39.
  • Medium: 3, 4, 5 × 8 gives 24, 32, 40; check 576 + 1024 = 1600.
  • Exam-type: Missing number in 20, ?, 52 (a multiple of 5, 12, 13): 52 ÷ 13 = 4, so ? = 12 × 4 = 48.

Rule 2: Make triples with m and n — Pick whole numbers m > n. Then m² − n², 2mn and m² + n² always form a triple.

  • Easy: m = 2, n = 1: 3, 4, 5.
  • Medium: m = 4, n = 3: 7, 24, 25.
  • Exam-type: m = 5, n = 2: 21, 20, 29; check 441 + 400 = 841 = 29².

Rule 3: Odd-number pattern — For an odd n, the triple is n, (n² − 1) ÷ 2, (n² + 1) ÷ 2. The last two numbers are consecutive.

Live figure

Baudhāyana triples

In the app you can move this figure with your finger.

Open in the app

In the figure: Type any three numbers and the checker tells you whether they make a right triangle.

  • Easy: n = 7: (49 − 1) ÷ 2 = 24, (49 + 1) ÷ 2 = 25, so 7, 24, 25.
  • Medium: n = 13: 84 and 85, so 13, 84, 85.
  • Exam-type: Is 16, 63, 65 a triple? 256 + 3969 = 4225 = 65². Yes.
Remember

All three numbers of a triple must be whole numbers. A triple 3, 4, 5 and its multiples (6, 8, 10), (9, 12, 15) are all right triangles of the same shape.

Common mistake

Adding the same number to each member (3 + 1, 4 + 1, 5 + 1) instead of multiplying. Only multiplying keeps it a triple.

Shortcut

Multiply to scale, "Minus, Twice, Plus" to build. Memory clue: m² Minus n², Twice mn, m² Plus n².

Spot the difference

PointScalingThe m, n rule
Start froma known tripletwo whole numbers m > n
Example3, 4, 5 × 3 = 9, 12, 15m = 3, n = 2: 5, 12, 13
Givesa bigger similar triplea brand new triple

Practice questions

Easy
  1. 1Multiply the triple 3, 4, 5 by 10.
  2. 2Multiply the triple 5, 12, 13 by 2.
  3. 3Use m = 2 and n = 1 in m² − n², 2mn, m² + n².
Medium
  1. 4Use m = 3 and n = 2 to make a triple.
  2. 5Use the odd number 15 to make a triple.
  3. 6Find the missing number in ?, 24, 25.
Challenging
  1. 7The triple 16, 30, 34 is a multiple of which smaller triple?
  2. 8Is 11, 60, 61 a triple?
Show answers (try them yourself first)
  1. 130, 40, 50
  2. 210, 24, 26
  3. 33, 4, 5
  4. 45, 12, 13
  5. 515, 112, 113
  6. 67
  7. 78, 15, 17 (× 2)
  8. 8Yes (121 + 3600 = 3721 = 61²)

Applications of the theorem

From daily life

A painter wants to reach a window 12 m up a wall using a 13 m ladder. Where should the foot of the ladder be placed? The theorem tells us exactly: 5 m from the wall.

Simple definition

Whenever a problem has a right angle (ladder and wall, rectangle and its diagonal, rhombus and its diagonals, a path that turns by 90°), draw the right triangle, mark the hypotenuse and use a² + b² = c².

13125

In the figure: Ladder 13 is the hypotenuse: 13² − 12² = 25, so the foot is 5 m away.

Rule 1: Ladder and wall — The ladder is the hypotenuse. Height² = ladder² − distance².

Live figure

Applications of the theorem

In the app you can move this figure with your finger.

Open in the app

In the figure: Change the ladder length and foot distance to see how high the ladder reaches.

  • Easy: Ladder 5 m, foot 3 m from the wall: height = 4 m.
  • Medium: Ladder 17 m, foot 8 m: height² = 289 − 64 = 225, height 15 m.
  • Exam-type: Tree breaks 12 m above ground and the tip touches the ground 9 m away: broken part = 15 m, total height 12 + 15 = 27 m.

Rule 2: Rectangle and its diagonal — Diagonal² = length² + breadth².

  • Easy: 12 m by 16 m plot: diagonal 20 m.
  • Medium: Diagonal 25 cm, one side 24 cm: other side 7 cm.
  • Exam-type: Diagonal 41 m, length 40 m: breadth 9 m, perimeter 98 m.

Rule 3: Rhombus and walking paths — Diagonals of a rhombus cross at right angles and bisect each other, so side² = (half of one diagonal)² + (half of the other)². For a path with a 90° turn, the straight-line distance is the hypotenuse.

  • Easy: Walk 9 m east, then 12 m north: straight distance 15 m.
  • Medium: Rhombus with diagonals 48 and 20: halves 24 and 10, side = 26 cm.
  • Exam-type: Rhombus with diagonals 16 and 30: halves 8 and 15, side = 17 cm, perimeter 68 cm.
Remember

Always draw a neat figure first and write the lengths on it. Check the answer: the hypotenuse must be longer than each of the other sides.

Common mistake

Using the half-diagonals in the rhombus problem as the full diagonals, or adding when you should subtract (finding a leg). Ladder, rope or diagonal = hypotenuse.

Shortcut

Spot the right angle, name the longest side. Memory clue: "Draw it, label it, square it".

Spot the difference

PointFinding the hypotenuseFinding a leg
Operationadd the squaressubtract the squares
Example9² + 12² = 22517² − 8² = 225
Thensquare root: 15square root: 15

Practice questions

Easy
  1. 1A 10 m ladder has its foot 6 m from a wall. How high does it reach?
  2. 2A plot is 12 m by 16 m. Find its diagonal.
  3. 3Asha walks 9 m east and 12 m north. How far is she from the start in a straight line?
Medium
  1. 4A 17 m ladder has its foot 8 m from a wall. How high does it reach?
  2. 5The diagonals of a rhombus are 48 cm and 20 cm. Find its side.
  3. 6A rectangle has diagonal 25 cm and one side 24 cm. Find the other side.
Challenging
  1. 7A tree breaks 12 m above the ground and the top touches the ground 5 m from the foot. Find the original height.
  2. 8A rectangle has diagonal 41 m and length 40 m. Find its perimeter.
Show answers (try them yourself first)
  1. 18 m
  2. 220 m
  3. 315 m
  4. 415 m
  5. 526 cm
  6. 67 cm
  7. 725 m
  8. 898 m