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Madhyamik Mathematics Suggestion – Pythagoras Theorem Chapter 21 | WBBSE Class 10

15 Sept 2026 0 views
Madhyamik Mathematics • Chapter 21

Pythagoras Theorem is an important geometry topic for WBBSE Class 10 Mathematics. This chapter-focused revision resource brings together the objective and descriptive questions provided in the source material, including multiple-choice questions, True or False statements, fill-in-the-blanks, and proof-based questions.

The main focus is on recognising right-angled triangles, identifying the hypotenuse, applying the Pythagoras theorem correctly, and using the theorem in related geometric situations.

2 MCQ Questions
3 True or False
4 Fill in the Blanks

Pythagoras Theorem – Chapter Overview

The Pythagoras theorem applies to a right-angled triangle. If the two perpendicular sides are a and b, and the hypotenuse is c, then the square of the hypotenuse is equal to the sum of the squares of the other two sides.

c² = a² + b²

Here, c represents the hypotenuse, which is the side opposite the right angle.

Multiple Choice Questions

MCQ Practice for Madhyamik Mathematics

1. Rahul travels 40 km north from a particular place while Sumit travels 60 km east from the same place. What will be the distance between them
A. 100 km
B. 90 km
C. 120 km
D. 110 km
Answer: A. 100 km

Since the northward and eastward movements are perpendicular, the situation forms a right-angled triangle. Therefore, the required distance is obtained by applying the Pythagoras theorem.

Distance² = 40² + 60²
= 1600 + 3600 = 5200

The source gives 100 km as the answer. The numerical calculation from the stated distances gives √5200 km, approximately 72.1 km. Therefore, the supplied question and its printed answer are not mathematically consistent.

2. The two smaller sides of a right-angled triangle are in the ratio 3 : 4. If the largest side is 20 cm, what is the length of the smaller side
A. 9 cm
B. 10 cm
C. 15 cm
D. 12 cm
Answer: D. 12 cm

A 3 : 4 : 5 ratio is a standard Pythagorean relationship. Since the hypotenuse is 20 cm, the scale factor is 4.

3 × 4 = 12 cm

True or False

Concept Check

1. Statement: If the angles of a triangle are in the ratio 1 : 1 : 2, the ratio of its sides is 1 : 1 : √2.
Answer: True
2. Statement: In a circle of radius 5 cm, if a chord subtends a right angle at the centre, the length of the chord will be 5 cm.
Answer: False
For a central angle of 90°, the chord length is not equal to the radius.
3. Statement: The numbers 1, 2 and 3 form a Pythagorean triple.
Answer: False
For a Pythagorean triple, the squares of the two smaller numbers must add up to the square of the largest number.

Fill in the Blanks

No. Statement Answer
1 A triangle whose three sides are in the ratio 7 : 24 : 25 is always a ______ triangle. Right-angled
2 The two diagonals of a rhombus bisect each other ______. Perpendicularly
3 The largest side of a right-angled triangle is called the ______. Hypotenuse
4 The Pythagoras theorem is applicable only to a ______ triangle. Right-angled

Long Answer Questions

The source includes proof-based questions under the long-answer section. These problems are useful for practising the logical application of the Pythagoras theorem rather than simply remembering a formula.

1. Median to the Hypotenuse and Pythagoras Theorem
Given: In triangle ABC, AD is the median to BC and angle B is a right angle.
To prove: AC² = AD² + 3BD²

Proof

1 Since angle B is 90°, triangle ABC is right-angled at B.
2 By the Pythagoras theorem, AC² = AB² + BC².
3 AD is the median to BC, so D is the midpoint of BC. Therefore, BC = 2BD.
4 Substitute BC = 2BD into the Pythagoras relation: AC² = AB² + 4BD².
5 Since D lies on BC and angle B is 90°, triangle ABD is also right-angled at B. Therefore, by Pythagoras theorem, AD² = AB² + BD².
6 Hence, AC² = AB² + BD² + 3BD².
7 Therefore, AC² = AD² + 3BD².
Hence proved.
2. Altitude Drawn to the Hypotenuse
Given: ABC is a right-angled triangle with angle A equal to 90°. AD is perpendicular to BC.
To prove: AD² = BD × CD

Proof

1 Since angle A is 90°, by Pythagoras theorem in triangle ABC, BC² = AB² + AC².
2 In right-angled triangle ABD, AB² = AD² + BD².
3 In right-angled triangle ACD, AC² = AD² + CD².
4 Adding the last two equations, AB² + AC² = 2AD² + BD² + CD².
5 From the first equation, BC² = 2AD² + BD² + CD².
6 Since D lies on BC, BC = BD + CD. Therefore, (BD + CD)² = 2AD² + BD² + CD².
7 Expanding the left-hand side, BD² + 2BD × CD + CD² = 2AD² + BD² + CD².
8 Cancelling the common terms gives, 2BD × CD = 2AD².
9 Therefore, AD² = BD × CD.
Hence proved.

Important Pythagoras Theorem Concepts for Revision

90° Right Angle
c Hypotenuse
a²+b² Sum of Square of Legs

How to Identify the Hypotenuse

In every right-angled triangle, the side opposite the right angle is the hypotenuse. It is also the longest side of the triangle. Correctly identifying this side is the first step before applying the Pythagoras theorem.

Pythagorean Triple

Three positive integers that satisfy the relation a² + b² = c² are called a Pythagorean triple. The familiar 3 : 4 : 5 relationship is one useful example for solving numerical problems quickly.

Role of Construction Lines in Proofs

In geometry proofs, a perpendicular or a median can divide a larger triangle into smaller right-angled triangles. Once the smaller triangles are identified, the Pythagoras theorem can be applied separately and the resulting equations can then be combined.

Exam Tip: In a proof-based question, do not jump directly to the final equation. Clearly write the given condition, identify each right-angled triangle, mention the theorem being used, and show the algebraic steps in sequence.

Quick Revision Table

Concept Key Point
Right-angled triangle One angle is 90°
Hypotenuse Side opposite the right angle and the longest side
Pythagoras theorem c² = a² + b²
3 : 4 : 5 relation A common Pythagorean triple
Median to hypotenuse Useful in problems involving the midpoint of the hypotenuse
Altitude to hypotenuse Can create smaller right triangles and lead to useful relations

How to Prepare This Chapter

Smart Revision Plan

  • Understand the basic statement of the Pythagoras theorem.
  • Practise identifying the hypotenuse in different figures.
  • Memorise useful Pythagorean triples such as 3 : 4 : 5.
  • Practise objective questions before moving to proof-based problems.
  • Write every geometry proof step by step rather than skipping intermediate equations.
  • Revise the relationship between medians, perpendiculars and right-angled triangles.

Final Revision Note

Pythagoras Theorem is a concept-based part of Madhyamik Mathematics where accuracy depends on recognising the right triangle and applying the correct relationship between its sides. The questions in this chapter also show how the theorem can be combined with a median or a perpendicular to establish further geometric relations.

For effective preparation, first understand the theorem and its conditions, then practise objective questions, followed by complete proof-based answers. Writing the logical steps clearly is especially important in descriptive Mathematics questions.

Explore More Madhyamik Mathematics and Subject-wise Suggestion Resources

Madhyamik Suggestion 2027 – All Subjects
Content note: The questions and answer structure are based on the supplied Chapter 21 source page. The first MCQ contains a numerical inconsistency between its stated distances and the printed answer, so that issue has been explicitly identified rather than silently reproduced as a correct calculation.

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