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Pythagoras Theorem — REET Maths Notes

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This page covers REET maths' 'Pythagoras Theorem' topic as per the syllabus — the statement of the theorem, common Pythagorean triplets, the converse theorem, and practical examples like ladders and poles are explained with self-checked calculations. This topic is included in the geometry section for both REET Level 1 and Level 2 maths papers, so knowing the triplets and correctly identifying the hypotenuse matters.

  • 0 questions in REET 2024 (syllabus-based notes)
  • Class 7–8 level
  • 8 min read
Verified 17 September 2026 · Source: BSER — REET आधिकारिक सिलेबस व 2024 प्रश्न-पत्र (मूल PDF; बोर्ड पोर्टल पर अब उपलब्ध नहीं)

How this topic is asked in REET

No direct question on 'Pythagoras Theorem' was found in the REET 2024 paper records, yet it is a core part of the geometry section (Class 7-8 level) of the official BSER maths syllabus. Right-angled triangle sides, Pythagorean triplets, and practical questions involving ladders and poles are common in competitive exams generally. The notes below are based entirely on the syllabus wording and self-checked calculations — no imagined 'previous question' has been used.

The statement and its history

According to the Pythagoras Theorem, in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides — that is, if the sides are a, b and the hypotenuse is c, then a² + b² = c². This describes a relationship between the sides of a triangle, not its angles, so it holds true only for right-angled triangles.

This relationship is famously named after the Greek mathematician Pythagoras (around 570-495 BCE), but the same relationship is also found described in the ancient Indian 'Baudhayana Shulba Sutra' (around 800 BCE) in the geometry used for altar construction — meaning it was known in India even before Pythagoras. That's exactly why NCERT textbooks also mention it under the alternative name 'Baudhayana theorem'.

Formulas worth knowing

Basic formula
a² + b² = c² (c = hypotenuse, a,b = the other two sides)
Finding the hypotenuse
c = √(a² + b²)
Finding one side
a = √(c² − b²)
The converse of the theorem
if a²+b² = c², the triangle is right-angled, with the right angle opposite side c
Diagonal of a rectangle
diagonal = √(length² + width²)

Common Pythagorean triplets

Triplet (a, b, c)Check (a²+b²=c²)
3, 4, 59 + 16 = 25 = 5²
5, 12, 1325 + 144 = 169 = 13²
6, 8, 1036 + 64 = 100 = 10²
8, 15, 1764 + 225 = 289 = 17²
7, 24, 2549 + 576 = 625 = 25²
9, 12, 1581 + 144 = 225 = 15²
20, 21, 29400 + 441 = 841 = 29²

Every triplet here is verified by squaring it out — recognising these directly in the exam saves time.

Solved practical examples

Example 1: a 13m long ladder is leaning against a wall, with its foot 5m away from the wall. How high up the wall does the ladder reach? Solution: height = √(13² − 5²) = √(169−25) = √144 = 12m.

Example 2: find the diagonal of a rectangular field 6m long and 8m wide. Solution: diagonal = √(6²+8²) = √(36+64) = √100 = 10m.

Example 3: a 24m tall pole is tied to the ground with a wire, with the other end of the wire fixed to the ground 7m from the base of the pole. Find the length of the wire. Solution: length of wire = √(24²+7²) = √(576+49) = √625 = 25m.

Examples using the converse theorem

Triangle's sidesIs it a right-angled triangle?
10cm, 24cm, 26cmYes — 10²+24² = 100+576 = 676 = 26² (hypotenuse 26cm)
5cm, 6cm, 7cmNo — 5²+6² = 25+36 = 61 ≠ 49 (7²)
9cm, 40cm, 41cmYes — 9²+40² = 81+1600 = 1681 = 41² (hypotenuse 41cm)

The converse theorem lets you check whether a triangle is right-angled without measuring any angle — checking the squares of the three sides is enough.

Use in direction-and-distance questions

The use of the Pythagoras Theorem isn't limited to ladder-and-pole type questions; when a person or object moves in two mutually perpendicular directions (say, first north, then east), the same theorem is used to find the straight-line (aerial) distance between the starting and final points, because two perpendicular directions together form a right-angled triangle.

Example: a person walks 15 km north from home, then 20 km east from there. Find the straight-line distance from home. Solution: north and east are mutually perpendicular directions, so the straight-line distance = √(15²+20²) = √(225+400) = √625 = 25 km (this is the 15-20-25 triplet, which is 5 times the 3-4-5 triplet).

REET papers typically build three kinds of questions from this topic — directly finding the hypotenuse or a side, identifying a right-angled triangle via the converse theorem, and practical (ladder, pole, direction) questions like the ones above. All three types rest on the same one formula, a²+b²=c², so knowing it well and recognising the common triplets is the key to solving them fast.

How to teach this to children

Activity

Draw a right-angled triangle on graph paper and build squares on all three sides, then have students count the unit squares inside each square to show that the count in the two smaller squares adds up to the count in the larger square — this 'cut and count' method is the most effective classroom activity for proving the theorem.

Teaching aid

Giving students coloured-paper squares cut in the proportions of a 3-4-5 triplet and having them rearrange the pieces themselves (a cut-and-rearrange proof) gives the theorem a concrete, tangible form.

Common misconception

Students often apply the theorem to any triangle, when it is true only for right-angled triangles; there's also confusion about identifying the hypotenuse — the hypotenuse is always the side directly opposite the right angle (and the longest side).

Assessment idea

Give students a triangle with three given sides and ask 'is this a right-angled triangle?', having them use the converse theorem to write a reasoned answer.

A trick worth remembering

First identify the longest of the three sides as the hypotenuse (c), then apply a²+b²=c². Memorising the common triplets (3-4-5, 5-12-13, 8-15-17, 7-24-25) and their multiples (like 6-8-10, 9-12-15) saves a lot of calculation time.

Practice questions

Question 1

In a right triangle, the two legs are 9cm and 12cm. Find the hypotenuse.

  1. A13cm
  2. B18cm
  3. C21cm
  4. D15cm

Explanation

Hypotenuse = √(9²+12²) = √(81+144) = √225 = 15cm (this is the 3-4-5 triplet scaled by 3, i.e. 9-12-15).

Question 2

In a right triangle, the hypotenuse is 17cm and one leg is 8cm. Find the other leg.

  1. A9cm
  2. B12cm
  3. C15cm
  4. D13cm

Explanation

The other side = √(17²−8²) = √(289−64) = √225 = 15cm (this is the common triplet 8-15-17).

Question 3

A 25m long ladder leans against a wall with its foot 7m from the wall. How high up the wall does the ladder reach?

  1. A18m
  2. B24m
  3. C20m
  4. D22m

Explanation

Height = √(25²−7²) = √(625−49) = √576 = 24m (triplet 7-24-25).

Question 4

The sides of a rectangle are 5cm and 12cm. Find its diagonal.

  1. A13cm
  2. B17cm
  3. C10cm
  4. D15cm

Explanation

Diagonal = √(5²+12²) = √(25+144) = √169 = 13cm.

Question 5

Which of the following is a valid Pythagorean triplet?

  1. A2, 3, 4
  2. B9, 10, 11
  3. C6, 8, 10
  4. D5, 6, 7

Explanation

6²+8² = 36+64 = 100 = 10², so this is a valid triplet (the 3-4-5 triplet scaled by 2). In every other option, a²+b² ≠ c².

Question 6

A relation similar to the Pythagoras theorem, predating Pythagoras, is found in which ancient Indian text?

  1. Aलीलावती
  2. Bबौधायन शुल्ब सूत्र
  3. Cगणित सार संग्रह
  4. Dआर्यभटीय

Explanation

The Baudhayana Shulba Sutra (around 800 BCE) predates Pythagoras (around 570-495 BCE) and describes the same relationship in the geometry of altar construction; the Aryabhatiya (5th century CE), Lilavati and Ganita Sara Sangraha are all much later works.

Question 7

A triangle has sides 10cm, 24cm and 26cm. Is it right-angled, and if so, which side is the hypotenuse?

  1. Aहाँ, कर्ण 10cm है
  2. Bनहीं, यह समकोण त्रिभुज नहीं है
  3. Cहाँ, कर्ण 24cm है
  4. Dहाँ, कर्ण 26cm है

Explanation

10²+24² = 100+576 = 676 = 26², so this is a right-angled triangle, and the longest side, 26cm, is the hypotenuse (by the converse theorem).

Question 8

A wire is tied from the top of a 10m pole to a point on the ground 24m from its base. Find the length of the wire.

  1. A26m
  2. B28m
  3. C25m
  4. D22m

Explanation

Length of wire = √(10²+24²) = √(100+576) = √676 = 26m (triplet 10-24-26, which is the 5-12-13 triplet scaled by 2).

Question 9

Which is the most suitable activity to practically explain the Pythagoras theorem in Class 7?

  1. Aकेवल प्रमेय का कथन बोर्ड पर लिखकर रटवाना
  2. Bग्राफ़ पेपर पर समकोण त्रिभुज की भुजाओं पर वर्ग बनाकर खाने गिनवाना और संबंध स्वयं निकलवाना
  3. Cव्यावहारिक प्रदर्शन को अनावश्यक मानकर छोड़ देना
  4. Dबिना किसी उदाहरण के सूत्र से सीधे प्रश्न हल करवाना

Explanation

Building squares on the sides of a right-angled triangle on graph paper and counting the unit squares lets students discover the relationship for themselves (discovery-based learning), which teaches far more effectively and durably than rote learning.

Question 10

What is the converse of the Pythagoras theorem used for?

  1. Aत्रिभुज का परिमाप ज्ञात करने के लिए
  2. Bत्रिभुज का क्षेत्रफल ज्ञात करने के लिए
  3. Cयह जाँचने के लिए कि दी गई भुजाओं वाला त्रिभुज समकोण है या नहीं
  4. Dत्रिभुज के कोणों का योग ज्ञात करने के लिए

Explanation

The converse theorem states that if a²+b²=c² holds for a triangle's sides, then the triangle is right-angled — this lets you check without measuring any angle.

Frequently asked questions

How many questions come from 'Pythagoras Theorem' in the REET exam?+

No direct question from this topic was found on record in REET 2024, but it is included in the geometry section of the official BSER maths syllabus, so preparing it is necessary.

What kind of triangles does the Pythagoras Theorem apply to?+

It applies only to right-angled triangles — not to any triangle.

What is a Pythagorean triplet?+

Three natural numbers that satisfy a²+b²=c², such as 3,4,5 or 5,12,13; these form the sides of a right-angled triangle, and memorising them speeds up calculation.

By what other name is the Pythagoras Theorem known in India?+

The same relationship is found described earlier in the Baudhayana Shulba Sutra (around 800 BCE), which predates Pythagoras (around 570-495 BCE), so it's also called the 'Baudhayana theorem' in several textbooks.

Is the Pythagoras Theorem also used in direction-and-distance questions?+

Yes, if two directions of motion are mutually perpendicular (like north and east), the Pythagoras Theorem itself is used to find the straight-line distance between the starting and final points.

Read next

Triangles and Congruence — REET Notes

Previous topic: read about the concept and formulas of triangles and congruence.

Quadrilaterals — REET Notes

Next topic: read about the concept and formulas of quadrilaterals.

REET Syllabus 2024

See the complete official REET Level 1 and Level 2 syllabus, subject-wise.

REET 2024 Solved Papers

Read the REET 2024 previous papers with answers and explanations.

Official Sources

  1. BSER — REET आधिकारिक सिलेबस व 2024 प्रश्न-पत्र (मूल PDF; बोर्ड पोर्टल पर अब उपलब्ध नहीं)
  2. NCERT Class 8 Maths (Ganita Prakash) Part II — Chapter 2: The Baudhāyana–Pythagoras Theorem

Change Log

  • 17 September 2026 — English page published — mirrors the fact-checked Hindi page, independently bilingual-accuracy checked