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09
NCERT Solutions · Maths Ch 9
The Baudhayana-Pythagoras Theorem
19 questions · 4 exercises
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Ex 9.1
Ex 9.2
Ex 9.3
Ex 9.4
NCERT answers with full exam-style working, step by step. Tap a question to open its solution.
1
EXERCISE 9.1 · 3 QUESTIONS
Doubling a square
Q1
Two identical square papers are cut as shown (pieces 1, 2 from one square and 3, 4 from the other). Can you arrange these pieces to create a square with double the area of either square?
Figure: each square is cut along a diagonal, giving four congruent right isosceles triangles.
(2 marks)
Q2
The length of the two equal sides of an isosceles right triangle is given. Find the length of the hypotenuse. Find bounds on the length of the hypotenuse such that they have at least one digit after the decimal point. (i) 3 (ii) 4 (iii) 6 (iv) 8 (v) 9
(5 marks)
Q3
The hypotenuse of an isosceles right triangle is 10. What are its other two side lengths?
(2 marks)
2
EXERCISE 9.2 · 4 QUESTIONS
Two different squares
Q1
If a right-angled triangle has shorter sides of lengths 5 cm and 12 cm, what is the length of its hypotenuse? Check using Baudhāyana's theorem.
(2 marks)
Q2
A right-angled triangle has a short side of 8 cm and hypotenuse of 17 cm. What is the third side?
(2 marks)
Q3
Using the constructions you have seen, how would you construct a square whose area is triple the area of a given square? Five times the area of a given square? (Baudhāyana's Śulba-Sūtra, Verse 1.10)
(3 marks)
Q4
Let a, b, c be the sides of a right triangle, c the hypotenuse. Find the missing side: (i) a = 5, b = 7 (ii) a = 8, b = 12 (iii) a = 9, c = 15 (iv) a = 7, b = 12 (v) a = 1.5, b = 3.5
(5 marks)
3
EXERCISE 9.3 · 3 QUESTIONS
Integer triples
Q1
Find 5 more Baudhāyana triples using the idea (1 + 3 + 5 + … + (2n − 1) = n², so (n − 1)² + (2n − 1) = n²) shown in the chapter.
(5 marks)
Q2
Does this method yield non-primitive Baudhāyana triples? [Hint: one of the smaller side lengths is one less than the hypotenuse.]
(2 marks)
Q3
Are there primitive triples that cannot be obtained through this method? If yes, give examples.
(2 marks)
4
EXERCISE 9.4 · 9 QUESTIONS
Applications
Q1
Find the diagonal of a square with side length 5 cm.
(2 marks)
Q2
Find the missing side lengths in the right triangles shown: (a) legs 7 and 9; (b) legs 4 and 10; (c) side 40 and hypotenuse 41; (d) leg 10 and hypotenuse √200; (e) leg 10 and hypotenuse √150; (f) leg 27 and hypotenuse 45.
Figure: six right-angled triangles with the right angle marked.
(5 marks)
Q3
Find the side length of a rhombus whose diagonals are of length 24 units and 70 units.
(2 marks)
Q4
Is the hypotenuse the longest side of a right triangle? Justify your answer.
(2 marks)
Q5
True or False — Every Baudhāyana triple is either a primitive triple or a scaled version of a primitive triple.
(2 marks)
Q6
Give 5 examples of rectangles whose side lengths and diagonals are all integers.
(3 marks)
Q7
Construct a square whose area is equal to the difference of the areas of squares of side lengths 5 units and 7 units.
(3 marks)
Q8
(i) Using the dots of a grid as the vertices, can you create a square of area (a) 2, (b) 3, (c) 4, (d) 5 sq. units? (ii) If the grid extends indefinitely, what are the possible integer-valued areas of squares you can create in this manner?
(2 marks)
Q9
Find the area of an equilateral triangle with side length 6 units. [Hint: Show that an altitude bisects the opposite side.]
(3 marks)