NCERT Solutions for Class 8 Mathematics Chapter 2 Baudhayanapythagoras Theorem
NCERT Solutions for Class 8 Mathematics Chapter 2 Baudhayanapythagoras Theorem
NCERT solutions for Class 8 Mathematics Chapter 2 Baudhāyana-Pythagoras Theorem – complete answers & explanations
In this blog, we explore the complete NCERT solutions for Class 8 Mathematics Chapter 2, titled "Baudhāyana-Pythagoras Theorem". This chapter dives deep into the fundamental geometry concept that relates the three sides of a right-angled triangle. The theorem is attributed to the Indian mathematician Baudhāyana, who stated it several centuries before Pythagoras. The chapter covers the application of this theorem, how to calculate the hypotenuse, and its connections to real-world geometry. Download the worksheet and practice alongside solutions for better clarity. Book a free trial now to get expert guidance.
What this NCERT chapter covers?
1. Introduction to Baudhāyana’s theorem and its historical significance.
2. Derivation of the Pythagorean theorem in right-angled triangles.
3. Methods for doubling and halving squares.
4. Finding the hypotenuse of right-angled triangles and isosceles right triangles.
5. Understanding and using Baudhāyana triples and their properties.
6. Application of the theorem to various geometric problems.
How to use these NCERT solutions?
1. Attempt each question independently: Start by solving the questions on your own. This helps develop problem-solving skills and a deeper understanding of the concepts.
2. Review the answers: After attempting the questions, use the provided solutions to compare your answers and identify any mistakes.
3. Ask for guidance: Parents and teachers can help explain difficult concepts such as Baudhāyana triples and the geometric proofs.
4. Follow the exact order: These solutions follow the worksheet’s question order for easy comparison and learning.
Important tips & tricks for students
1. Grasp the fundamental theorem: Understand the relation between the sides of a right triangle using Baudhāyana’s theorem, \( a^2 + b^2 = c^2 \), where \( a \) and \( b \) are the legs, and \( c \) is the hypotenuse.
2. Master calculating the hypotenuse: When given the sides of a right triangle, use the Pythagorean theorem to find the hypotenuse.
3. Understand the significance of Baudhāyana triples**: Recognize the patterns of integer sides of right triangles that satisfy the Pythagorean theorem, such as (3, 4, 5), (5, 12, 13), and (7, 24, 25).
4. Use geometry constructions: Practice constructing geometric shapes, like squares and right triangles, to understand how the theorem applies visually.
5. Solve real-world problems: Apply the theorem to practical problems, such as calculating the depth of a lake or finding the dimensions of geometric shapes.
NCERT solutions – complete answer key
1. Doubling a Square
- Baudhāyana’s Śulba-Sūtra provides a construction method for doubling the area of a square by constructing a square on its diagonal.
- Answer: The area of the new square is twice the area of the original square because it consists of four congruent right-angled triangles.
2. Halving a Square
- To halve the area of a square, Baudhāyana suggests constructing a smaller square inside the larger square, using perpendicular lines for clarity.
- Answer: The smaller square has half the area because the total area is divided into two equal parts by the diagonal.
3. Hypotenuse of an Isosceles Right Triangle
- In an isosceles right triangle, the hypotenuse \( c \) is found using the formula \( a^2 + a^2 = c^2 \).
- Answer: If the sides \( a = 12 \), then \( c = \sqrt{288} \), and the hypotenuse is approximately between 16 and 17 units.
4. Using Baudhāyana’s Theorem to find the hypotenuse
- Example 1: For a right triangle with sides 3 cm and 4 cm, the hypotenuse is 5 cm, based on the theorem.
- Answer: \( 3^2 + 4^2 = 5^2 \), confirming the hypotenuse is 5 cm.
5. Baudhāyana Triples
- Example: The triple (3, 4, 5) is a classic Baudhāyana triple, and multiples of it, like (6, 8, 10) and (9, 12, 15), are also valid triples.
- Answer: Such triples can be scaled by multiplying each term by any positive integer.
6. Combining Two Different Squares
- To combine two squares with different sizes, Baudhāyana uses the concept of constructing a right triangle and a square on its hypotenuse.
- Answer: The area of the square on the hypotenuse is the sum of the areas of the two smaller squares.
7. Right Triangles Having Integer Sidelengths
- Example: The list of Baudhāyana triples includes (3, 4, 5), (6, 8, 10), and (5, 12, 13). These triples satisfy the Pythagorean theorem.
- Answer: These triples are primitive if they have no common factors other than 1, and scaled versions can be formed by multiplying each term by a positive integer.
8. Applications of the Theorem
- Example: A lotus flower’s stem problem from Bhāskarāchārya’s Līlāvatī uses Baudhāyana’s theorem to determine the depth of a lake.
- Answer: The depth is found by solving the equation \( 3^2 + x^2 = (x + 1)^2 \), resulting in a depth of 4 units.
Why NCERT solutions help students?
These NCERT solutions are designed to provide clear, reliable, and step-by-step answers to problems based on Baudhāyana’s theorem. By practicing these solutions, students can build a solid foundation in geometry, specifically right-angled triangles and their properties. This understanding will boost students' confidence in solving mathematical problems and applying geometric concepts in real life.
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