Introduction to Pythagoras Theorem
The Pythagoras Theorem (পিথাগোরাসের উপপাদ্য) is a fundamental concept in Geometry, forming a bridge between Algebra and Geometry through the study of right-angled triangles. This theorem is essential in the Class 10 West Bengal Board Mathematics syllabus and is frequently used in both theoretical and practical mathematical problems. Understanding its derivation, applications, and proofs is crucial for mastering geometry and preparing for board examinations.
Statement of the Pythagoras Theorem
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Mathematical Expression:
If ABC is a right-angled triangle with ∠B = 90°, then:
AC2 = AB2 + BC2
Here,
- AC is the hypotenuse (অতিভুজ),
- AB and BC are the two perpendicular sides (লম্ব ও ভুমিক বাহু).
Proof of the Theorem (উপপাদ্যের প্রমাণ)
The proof of the Pythagoras Theorem as per WBBSE syllabus is based on geometric methods. Consider a right-angled triangle ABC where ∠B = 90°.
Construction:
Draw triangle ABC with right angle at B. Construct a square using the triangle on each side, and analyze the area of each square geometrically.
Explanation:
Using the area of squares and rearrangement, it can be shown that the area of the square on the hypotenuse equals the combined area of the squares on the other two sides.
Thus,
AC2 = AB2 + BC2
This geometric method gives a visual and logical understanding of the theorem and complies with the standards of school geometry.
Converse of Pythagoras Theorem (পিথাগোরাস উপপাদ্যের বিপরীত)
Statement:
If in a triangle, the square of one side is equal to the sum of the squares of the other two sides, then the triangle is a right-angled triangle.
Mathematical Expression:
If in triangle ABC,
AC2 = AB2 + BC2
then ∠B = 90°, and triangle ABC is right-angled at B.
This converse is used to determine whether a given triangle is right-angled, based on the lengths of the sides.
Applications of Pythagoras Theorem in Real Life and Mathematics
The Pythagoras Theorem is not just a theoretical concept but is widely used in real-life applications and various mathematical problems.
1. Finding the Length of a Side
If two sides of a right triangle are known, the third side can be calculated using the formula.
Example:
In triangle XYZ, right-angled at Y,
XY = 6 cm, YZ = 8 cm
Then, hypotenuse XZ can be calculated as:
XZ2=XY2+YZ2=62+82=36+64=100⇒XZ=√100=10 cm
- Verifying Right-Angled Triangles
The theorem is used to check whether a triangle with known side lengths is a right triangle.
Example:
Given sides of a triangle: 5 cm, 12 cm, and 13 cm
Check if it is right-angled:
52 + 122 = 25 + 144 = 169132 = 169 ⇒ 52 + 122 = 132 ⇒ Right-angled triangle
3. Practical Use Cases
- Measuring the height of a building using distance and shadow length.
- In construction and carpentry to ensure right angles.
- In navigation and GPS technology for calculating shortest paths.
Common Questions Based on the Chapter
The West Bengal Board includes a variety of question types from this chapter:
1. Numerical Problems
- Find the length of a missing side.
- Apply the theorem to solve real-life measurement problems.
2. Proof-based Questions
- Prove the theorem using given diagrams.
- Provide geometric construction and derive the result.
3. Application of Converse
- Determine whether a given triangle is right-angled using side lengths.
Important Notes (মূল ধারণাগুলি)
- The theorem is only valid in right-angled triangles.
- The hypotenuse is always opposite the right angle and is the longest side.
- The converse helps in triangle classification.
Conclusion
The Pythagoras Theorem is a cornerstone of Class 10 Mathematics under the West Bengal Board. A clear understanding of the theorem, its converse, and practical applications is vital for mastering geometry and solving real-world mathematical problems. This chapter provides both conceptual clarity and problem-solving strategies that are important for academic success and daily use of mathematics.