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State Board · Class 10 · Mathematics Part 2

Lesson 2: Pythagoras Theorem

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

This lesson is also in: मराठी

Pythagorean triplet

From daily life

A carpenter checks a corner with a 3-4-5 rope trick: 3 units one way, 4 units the other, and the gap is exactly 5. Numbers like 3, 4, 5 that fit the right-triangle rule are special.

Simple definition

A Pythagorean triplet (पायथागोरस त्रिकूट) is three natural numbers where the square of the largest equals the sum of the squares of the other two.

Rule 1: Checking a triplet — Square all three, then test largest² = sum of the other two squares. (9, 40, 41): 81 + 1600 = 1681 = 41², yes. (6, 7, 9): 36 + 49 = 85 but 9² = 81, no. The order of the numbers does not matter. Multiples also work: (3, 4, 5) × 3 = (9, 12, 15), and 81 + 144 = 225.

Squares on the sides 6, 8 and 10 of a right triangle with areas 36, 64 and 10036641006810Square areas:36 + 64 = 1006² + 8² = 10²(6, 8, 10) is aPythagorean triplet

In the figure : In the figure the squares on the two shorter sides add up to the square on the hypotenuse (6, 8, 10).

  • Easy : Easy: (20, 21, 29): 400 + 441 = 841 = 29², a triplet.
  • Medium : Medium: (9, 12, 16): 81 + 144 = 225 but 16² = 256, not a triplet.
  • Exam-type : Exam: The date 12/05/13 gives (12, 5, 13): 144 + 25 = 169 = 13², a triplet.

Rule 2: Formula for triplets — For natural numbers a > b, (a2 + b2, a2 − b2, 2ab) is a triplet. a = 4, b = 1: (17, 15, 8), and 225 + 64 = 289. a = 3, b = 2: (13, 5, 12). a = 6, b = 1: (37, 35, 12), and 1225 + 144 = 1369 = 37².

  • Easy : Easy: a = 2, b = 1 gives (5, 3, 4).
  • Medium : Medium: a = 5, b = 2 gives (29, 21, 20).
  • Exam-type : Exam: a = 7, b = 1 gives (50, 48, 14); check 2304 + 196 = 2500.
Remember

Square the largest first. Largest² = sum of the other two squares.

Common mistake

Adding the squares of the largest and a smaller number. The test uses the two smaller ones.

Shortcut

Memory clue: "लहान² + लहान² = मोठी²".

Spot the difference

TripletCheckResult
(9, 40, 41)81 + 1600 = 1681Yes
(6, 7, 9)36 + 49 ≠ 81No
(9, 12, 15)81 + 144 = 225Yes

Practice questions

Easy
  1. 1What is a Pythagorean triplet?
  2. 2Is (8, 15, 17) a triplet?
  3. 3Is (5, 6, 8) a triplet?
Medium
  1. 4Does the order matter?
  2. 5Make a triplet from (3, 4, 5) by multiplying by 4.
  3. 6Use a = 5, b = 1 in the formula.
Challenging
  1. 7Is the date 20/21/29 a triplet? Check.
  2. 8Prove (a2 + b2)2 = (a2 − b2)2 + (2ab)2.
Show answers (try them yourself first)
  1. 1Three natural numbers with largest² = sum of the other two squares.
  2. 264 + 225 = 289 = 17², yes.
  3. 325 + 36 = 61, 8² = 64, no.
  4. 4No, the largest is always tested.
  5. 5(12, 16, 20).
  6. 6(26, 24, 10); 576 + 100 = 676.
  7. 7400 + 441 = 841 = 29², yes.
  8. 8RHS = a4 − 2a2b2 + b4 + 4a2b2 = a4 + 2a2b2 + b4 = LHS.

Similarity of right angled triangles and theorem of geometric mean

From daily life

A plank rests on a wall, making a right angle. If you drop a perpendicular from the corner to the slanted plank, it splits the shape into two smaller triangles that look just like the big one, only smaller.

Simple definition

In a right triangle the altitude on the hypotenuse divides it into two triangles similar (समरूप) to the whole and to each other. Also altitude² = product of the two parts (geometric mean, भूमितीय मध्य).

Rule 1: Similar triangles — In ΔABC with ∠B = 90° and BD ⊥ AC, ΔADB ~ ΔABC (common ∠A, and 90° each) and ΔBDC ~ ΔABC (common ∠C). So ΔADB ~ ΔBDC as well. All three have the same shape.

A right triangle with the altitude BD on the hypotenuse AC: BD squared equals AD times DCACDBAD = 4DC = 9BD = 6BD² = AD × DC = 4 × 9 = 36

In the figure : In the figure the altitude on the hypotenuse is the geometric mean of the two parts: BD² = AD × DC.

  • Easy : Easy: The three triangles ΔADB, ΔBDC and ΔABC all have a right angle.
  • Medium : Medium: From the similarity, BDAD = DCBD, so BD² = AD × DC.
  • Exam-type : Exam: Why is ΔADB ~ ΔBDC? Both are similar to ΔABC, so by transitivity they are similar to each other.

Rule 2: Geometric mean theorem — Altitude² = (part 1) × (part 2). If the parts are 4 and 25, altitude = √100 = 10. If PS = 16 and SR = 9, then QS = √144 = 12. If QS = 6 and PS = 4, then SR = 364 = 9.

  • Easy : Easy: Parts 2 and 18: altitude = √36 = 6.
  • Medium : Medium: Parts 5 and 20: altitude = √100 = 10, and hypotenuse = 25.
  • Exam-type : Exam: AD = 5, DC = 15, so AC = 20. AB² = AD × AC = 100, so AB = 10. BC² = DC × AC = 300, and 100 + 300 = 400 = AC².
Remember

Altitude² = product of the two parts of the hypotenuse. The altitude is from the right angle.

Common mistake

Using the product formula for any altitude. It works only for the altitude on the hypotenuse of a right triangle.

Shortcut

Memory clue: "लंब² = भाग × भाग".

Spot the difference

GivenFindAnswer
Parts 4 and 25Altitude10
PS = 16, SR = 9QS12
QS = 6, PS = 4SR9

Practice questions

Easy
  1. 1Name the three similar triangles.
  2. 2State the geometric mean theorem.
  3. 3Parts 1 and 49: altitude?
Medium
  1. 4Parts 3 and 12: altitude?
  2. 5Altitude 8, one part 4. Other part?
  3. 6Altitude 12, parts in the hypotenuse sum to 25. Find the parts.
Challenging
  1. 7Parts 9 and 16: find both legs.
  2. 8Why is the altitude from the right angle the geometric mean?
Show answers (try them yourself first)
  1. 1ΔADB, ΔBDC and ΔABC.
  2. 2Altitude² = product of the two parts of the hypotenuse.
  3. 3√49 = 7.
  4. 4√36 = 6.
  5. 5644 = 16.
  6. 6x(25 − x) = 144, so 9 and 16.
  7. 79 × 25 = 225 gives 15; 16 × 25 = 400 gives 20.
  8. 8Similar triangles give BDAD = DCBD.

Pythagoras theorem and its converse

From daily life

A 13 m ladder leans on a wall with its foot 5 m away. How high does it reach? A right triangle hides here, and Pythagoras answers without a measuring tape.

Simple definition

Pythagoras theorem: in a right triangle, (hypotenuse)2 = (side)2 + (side)2. Its converse (विधानाचा विरुद्ध; converse): if c2 = a2 + b2 in a triangle, it is right angled.

Rule 1: Using the theorem — Legs 20 and 21: hypotenuse = √(400 + 441) = √841 = 29. Rectangle 24 by 7: diagonal = √(576 + 49) = √625 = 25. Ladder 13 m, foot 5 m: height = √(169 − 25) = √144 = 12 m.

A 10 m ladder leaning on an 8 m high wall has its foot 6 m from the wall8 m6 m ?ladder 10 mfoot² = 10² − 8²= 100 − 64 = 36foot = 6 mwall

In the figure : In the figure Pythagoras theorem applied to a ladder against a wall.

Live figure

Pythagoras theorem and its converse

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

Open in the app
  • Easy : Easy: Legs 6 and 8 give hypotenuse 10.
  • Medium : Medium: A ladder of 10 m with its top at 8 m height has its foot 6 m from the wall.
  • Exam-type : Exam: Two friends walk, one 9 km East and one 12 km North. Distance apart = √(81 + 144) = √225 = 15 km.

Rule 2: Converse: is it a right triangle? — Test the longest side. Sides 8, 15, 17: 64 + 225 = 289 = 17², right angled. Sides 9, 10, 13: 81 + 100 = 181 but 169, not right angled. Sides √5, √11, 4: 5 + 11 = 16, so the right angle is opposite the side 4.

  • Easy : Easy: 5, 12, 13 form a right triangle (25 + 144 = 169).
  • Medium : Medium: 7, 8, 11: 49 + 64 = 113 but 121, not right.
  • Exam-type : Exam: The right angle is always opposite the longest side.
Remember

Hypotenuse is opposite the right angle and is the longest side.

Common mistake

Squaring and adding when the hypotenuse is given. Then subtract.

Shortcut

Memory clue: "कर्ण² = बाजू² + बाजू²".

Spot the difference

SituationCalculationAnswer
Legs 20, 21√(400 + 441)29
Ladder 13, foot 5√(169 − 25)12 m
Sides 9, 10, 1381 + 100 ≠ 169Not right

Practice questions

Easy
  1. 1State Pythagoras theorem.
  2. 2State its converse.
  3. 3Legs 9 and 12: hypotenuse?
Medium
  1. 4Hypotenuse 17, one leg 8: other leg?
  2. 5Is 10, 24, 26 a right triangle?
  3. 6Rectangle length 16 cm, area 192 sq.cm: diagonal?
Challenging
  1. 7A 10 m ladder slides so the top is at 7 m height. How far is the foot from the wall?
  2. 8In ΔABC, AD ⊥ BC. Show AB² + CD² = BD² + AC².
Show answers (try them yourself first)
  1. 1In a right triangle, hypotenuse² = sum of the squares of the other two sides.
  2. 2If c² = a² + b², the triangle is right angled.
  3. 315.
  4. 4√(289 − 64) = 15.
  5. 5100 + 576 = 676 = 26², yes.
  6. 6Breadth 12, diagonal 20 cm.
  7. 7√(100 − 49) = √51 m, about 7.14 m.
  8. 8AD² = AB² − BD² = AC² − CD², rearrange.

Triangles of 30°-60°-90° and 45°-45°-90°

From daily life

A set square from your geometry box is a 30°-60°-90° or 45°-45°-90° triangle. Their sides always keep fixed ratios, so one side tells you the other two.

Simple definition

In a 30°-60°-90° triangle the sides are in the ratio 1 : √3 : 2; in a 45°-45°-90° triangle they are in the ratio 1 : 1 : √2.

Rule 1: 30°-60°-90° theorem — Side opposite 30° = 12 × hypotenuse. Side opposite 60° = √32 × hypotenuse. Hypotenuse 20: short side 10, long side 10√3. Short side 5: hypotenuse 10, long side 5√3. Converse: if a leg is half the hypotenuse, its opposite angle is 30°.

A 30-60-90 triangle with sides 5, 5 root 3 and 10 and a 45-45-90 triangle with sides 5 root 2, 5 root 2 and 1055√31030°60°30° – 60° – 90°5√25√21045°45°45° – 45° – 90°Hypotenuse 10 in both triangles

In the figure : In the figure special right triangles with hypotenuse 10: half and half × √3, or each leg 1/√2 of the hypotenuse.

  • Easy : Easy: Hypotenuse 12 gives short side 6 and long side 6√3.
  • Medium : Medium: Long side 9√3 means hypotenuse 18 and short side 9.
  • Exam-type : Exam: Height of an equilateral triangle of side 8 is √32 × 8 = 4√3.

Rule 2: 45°-45°-90° theorem — Each leg = 1√2 × hypotenuse. Hypotenuse 4√2: each leg 4. Square with diagonal 6√2: side 6, perimeter 24. Leg x: hypotenuse x√2.

  • Easy : Easy: Leg 5 gives hypotenuse 5√2.
  • Medium : Medium: Diagonal 14 gives side 14√2 = 7√2.
  • Exam-type : Exam: In ΔABC, AD ⊥ BC, ∠B = 60°, ∠C = 45°, AB = 12. ΔABD gives BD = 6, AD = 6√3. ΔADC gives DC = 6√3 and AC = 6√6. So BC = 6 + 6√3.
Remember

30° is opposite the shortest side. 1 : √3 : 2 and 1 : 1 : √2.

Common mistake

Putting √3 on the wrong side. The shortest side is opposite 30°, the middle side (with √3) opposite 60°.

Shortcut

Memory clue: "30-60-90 → 1:√3:2; 45-45-90 → 1:1:√2".

Spot the difference

TriangleRatio
30°-60°-90°1 : √3 : 2
45°-45°-90°1 : 1 : √2
Equilateral, side sHeight

Practice questions

Easy
  1. 1Ratio of sides in a 30°-60°-90° triangle?
  2. 2Ratio of sides in a 45°-45°-90° triangle?
  3. 3Hypotenuse 16 in a 30°-60°-90° triangle: shorter side?
Medium
  1. 4Leg 7 in a 45°-45°-90° triangle: hypotenuse?
  2. 5Height of an equilateral triangle of side 10?
  3. 6Square with diagonal 12√2: side and perimeter?
Challenging
  1. 7ΔRST: ∠S = 90°, ∠T = 30°, RT = 20. Find RS and ST.
  2. 8An equilateral triangle has height 3√3. Find its side and perimeter.
Show answers (try them yourself first)
  1. 11 : √3 : 2.
  2. 21 : 1 : √2.
  3. 38.
  4. 47√2.
  5. 55√3.
  6. 6Side 12, perimeter 48.
  7. 7RS = 10, ST = 10√3.
  8. 8√32s = 3√3, s = 6, perimeter 18.

Application of Pythagoras theorem and Apollonius theorem

From daily life

To find the length of a median (the line from a vertex to the midpoint of the opposite side) without drawing the triangle exactly, mathematicians use a neat relation named after Apollonius.

Simple definition

Apollonius theorem (अपोलोनियस प्रमेय): if M is the midpoint of BC in ΔABC, then AB2 + AC2 = 2AM2 + 2BM2.

Rule 1: Acute and obtuse angle relations — If ∠C is acute and AD ⊥ BC, then AB2 = BC2 + AC2 − 2·BC·DC. In the 13-14-15 triangle (BC = 14, AC = 13, AB = 15): 225 = 196 + 169 − 28·DC, so DC = 5. If ∠ACB is obtuse (D beyond C): AB2 = BC2 + AC2 + 2·BC·CD. With BC = 3, AC = 5, CD = 4: AB2 = 9 + 25 + 24 = 58.

  • Easy : Easy: Acute case: BC = 14, AC = 13, DC = 5 gives AB2 = 225.
  • Medium : Medium: Check by right triangles: AD2 = 169 − 25 = 144, AD = 12; BD = 9; 144 + 81 = 225.
  • Exam-type : Exam: Obtuse case: AD = 3, BD = 7 gives AB2 = 9 + 49 = 58.

Rule 2: Using the Apollonius theorem — In ΔABC, BC = 12, AM = 5 gives BM = 6, so AB2 + AC2 = 50 + 72 = 122. If AB2 + AC2 = 178 and BC = 16: 178 = 2AM2 + 128, AM = 5. Sides 7, 9, 8: median to the side 8 has 130 = 2AM2 + 32, AM = 7. In a parallelogram the sum of the squares of the diagonals = the sum of the squares of all four sides.

A triangle with median AM of 5 and BM of 4 showing AB squared plus AC squared equals 82ABCMAM = 5BM = 4MC = 4Apollonius: AB² + AC² = 2AM² + 2BM²65 + 17 = 2 × 25 + 2 × 16 = 82

In the figure : In the figure the Apollonius theorem checked with numbers.

  • Easy : Easy: AM = 5, BM = 6 gives AB2 + AC2 = 122.
  • Medium : Medium: Parallelogram with adjacent side squares adding to 85 and one diagonal 11: 2 × 85 = 170 = 121 + d2, so d = 7.
  • Exam-type : Exam: In a rhombus, the sum of the squares of the diagonals equals 4 × (side)2.
Remember

AB² + AC² = 2AM² + 2BM². M is the midpoint, so BM = half of BC.

Common mistake

Using BC instead of BM. The formula has BM, half of BC.

Shortcut

Memory clue: "दोन बाजू² = 2 मध्यगा² + 2 अर्धा पाया²".

Spot the difference

GivenFindAnswer
BC = 12, AM = 5AB² + AC²122
AB² + AC² = 178, BC = 16AM5
Sides 7, 9, 8Median to 87

Practice questions

Easy
  1. 1State the Apollonius theorem.
  2. 2What is a median?
  3. 3AM = 4, BC = 10: AB² + AC²?
Medium
  1. 4AB² + AC² = 100, BC = 12: AM?
  2. 5Parallelogram sides 6 and 8, one diagonal 10: other diagonal?
  3. 6ΔABC: AB = 10, AC = 6, BC = 8. Find the median AM.
Challenging
  1. 7AB = AC = 13, BC = 10. Find AM and the area.
  2. 8Show that in a rhombus d12 + d22 = 4s2.
Show answers (try them yourself first)
  1. 1AB² + AC² = 2AM² + 2BM².
  2. 2A line from a vertex to the midpoint of the opposite side.
  3. 32 × 16 + 2 × 25 = 82.
  4. 4100 = 2AM² + 72, AM² = 14, AM = √14.
  5. 5200 = 100 + d², d = 10.
  6. 6136 = 2AM² + 32, AM² = 52, AM = 2√13.
  7. 7338 = 2AM² + 50, AM = 12; area = 12 × 10 × 12 = 60.
  8. 8Diagonals bisect at right angles, so (d1/2)² + (d2/2)² = s².

Quick revision

  • Pythagorean triplet: the square of the largest number is the sum of the squares of the other two, as in (3, 4, 5), and (a² + b², a² − b², 2ab) is a Pythagorean triplet. The altitude on the hypotenuse of a right angled triangle makes two triangles similar to each other and to the whole triangle, and the square of the altitude is the product of the two parts of the hypotenuse.
  • Pythagoras theorem: the square of the hypotenuse is the sum of the squares of the other two sides, and by its converse a triangle is right angled when the square of one side equals the sum of the squares of the other two. In a 30°-60°-90° triangle the sides are in the ratio 1 : √3 : 2, and in a 45°-45°-90° triangle they are in the ratio 1 : 1 : √2.
  • Apollonius theorem: AB² + AC² = 2AM² + 2BM², where M is the midpoint of BC; in a parallelogram the sum of the squares of the diagonals equals the sum of the squares of the sides.