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All questions of The BAUDHĀYANA- Pythagoras Theorem for Class 8 Exam
The diagonal of a square is 10 cm. What is the area of the square?- a)50 cm²
- b)100 cm²
- c)200 cm²
- d)25 cm²
Correct answer is option 'A'. Can you explain this answer?
a)
50 cm²
b)
100 cm²
c)
200 cm²
d)
25 cm²
 | Ayush Unni answered |
Understanding the Square's Diagonal
The diagonal of a square is a line segment that connects two opposite corners. In this case, the diagonal measures 10 cm.
Relationship Between Diagonal and Side Length
To find the area of the square, we first need to determine the length of one side (s) using the relationship between the diagonal (d) and the side length:
- The formula for the diagonal of a square is given by:
- d = s√2
- Rearranging this formula to find the side length:
- s = d / √2
Calculating the Side Length
Given that the diagonal is 10 cm, we can substitute this value into the formula:
- s = 10 / √2
- s = 10 / 1.414 (approximately)
- s ≈ 7.07 cm
Finding the Area of the Square
Now that we have the side length, we can calculate the area (A) of the square using the formula:
- A = s²
Substituting the side length we calculated:
- A ≈ (7.07)²
- A ≈ 50 cm²
Conclusion
Therefore, the area of the square is approximately 50 cm², which corresponds to option 'A'.
The diagonal of a square is a line segment that connects two opposite corners. In this case, the diagonal measures 10 cm.
Relationship Between Diagonal and Side Length
To find the area of the square, we first need to determine the length of one side (s) using the relationship between the diagonal (d) and the side length:
- The formula for the diagonal of a square is given by:
- d = s√2
- Rearranging this formula to find the side length:
- s = d / √2
Calculating the Side Length
Given that the diagonal is 10 cm, we can substitute this value into the formula:
- s = 10 / √2
- s = 10 / 1.414 (approximately)
- s ≈ 7.07 cm
Finding the Area of the Square
Now that we have the side length, we can calculate the area (A) of the square using the formula:
- A = s²
Substituting the side length we calculated:
- A ≈ (7.07)²
- A ≈ 50 cm²
Conclusion
Therefore, the area of the square is approximately 50 cm², which corresponds to option 'A'.
If (a, b, c) is a Pythagorean triple, which of the following is also a Pythagorean triple?- a)(a+1, b+1, c+1)
- b)(2a, 2b, 2c)
- c)(a², b², c²)
- d)(a, b, c²)
Correct answer is option 'B'. Can you explain this answer?
a)
(a+1, b+1, c+1)
b)
(2a, 2b, 2c)
c)
(a², b², c²)
d)
(a, b, c²)
 | Nidhi Bhatt answered |
Given (a,b,c) is a Pythagorean triple, a2 + b2 = c2.
For option B: (2a)2 + (2b)2 = 4a2 + 4b2 = 4(a2 + b2) = 4c2 = (2c)2. Hence (2a, 2b, 2c) is a Pythagorean triple.
The other options do not hold in general: option A would require (a+1)2 + (b+1)2 = (c+1)2, which imposes 2(a+b+1)=2c+1 and is not generally true. Option C would need a4 + b4 = c4, which is false in general. Option D replaces c by c2, giving c2 ≠ (c2)2 except in trivial cases; so it is not generally a Pythagorean triple.
If the sides of a right triangle are 9 cm and 12 cm, its hypotenuse is:- a)13 cm
- b)15 cm
- c)18 cm
- d)21 cm
Correct answer is option 'B'. Can you explain this answer?
a)
13 cm
b)
15 cm
c)
18 cm
d)
21 cm
| | Nidhi Bhatt answered |
Use the Pythagorean theorem: c = √(a2 + b2).
Compute: 92 = 81 and 122 = 144.
Sum = 81 + 144 = 225, so c = √225 = 15 cm.
Answer: 15 cm (option B).
The value of √2 lies between:- a)1.3 and 1.4
- b)1.41 and 1.42
- c)1.5 and 1.6
- d)1.2 and 1.3
Correct answer is option 'B'. Can you explain this answer?
a)
1.3 and 1.4
b)
1.41 and 1.42
c)
1.5 and 1.6
d)
1.2 and 1.3
| | Nidhi Bhatt answered |
Square the endpoints to locate √2.
1.412 = 1.9881 < 2.
1.422 = 2.0164 > 2.
Therefore √2 lies between 1.41 and 1.42.
Which of the following is a primitive Pythagorean triple?- a)(6, 8, 10)
- b)(9, 12, 15)
- c)(5, 12, 13)
- d)(12, 16, 20)
Correct answer is option 'C'. Can you explain this answer?
a)
(6, 8, 10)
b)
(9, 12, 15)
c)
(5, 12, 13)
d)
(12, 16, 20)
| | Nidhi Bhatt answered |
Definition: A primitive Pythagorean triple consists of positive integers a, b, c with a2 + b2 = c2 and gcd(a,b,c) = 1.
Check (5, 12, 13): 52 + 122 = 25 + 144 = 169 = 132.
gcd(5,12,13) = 1, so the triple is primitive.
Therefore the correct answer is (5, 12, 13) (Option C). The other given triples are scalar multiples of (3,4,5) and hence not primitive.
If the area of a square is doubled, the ratio of the new side to the original side is:- a)2 : 1
- b)√2 : 1
- c)1 : √2
- d)4 : 1
Correct answer is option 'B'. Can you explain this answer?
a)
2 : 1
b)
√2 : 1
c)
1 : √2
d)
4 : 1
| | Nidhi Bhatt answered |
Let the original side be a. Then the original area = a2.
New area = 2a2, so new side = √(2a2) = a√2.
Ratio of new side to original side = a√2 : a = √2 : 1.
The area of the square on the hypotenuse of a triangle with sides 6 cm and 8 cm is:- a)100 cm²
- b)144 cm²
- c)196 cm²
- d)256 cm²
Correct answer is option 'A'. Can you explain this answer?
a)
100 cm²
b)
144 cm²
c)
196 cm²
d)
256 cm²
| | Nidhi Bhatt answered |
Use the Pythagorean theorem for a right triangle: c2 = a2 + b2.
Here a = 6 cm and b = 8 cm, so c2 = 62 + 82 = 36 + 64 = 100, giving c = 10 cm.
The area of the square on the hypotenuse equals c2 = 100 cm2.
The decimal expansion of √2 is:- a)Terminating
- b)Repeating
- c)Non-terminating and non-repeating
- d)Rational
Correct answer is option 'C'. Can you explain this answer?
a)
Terminating
b)
Repeating
c)
Non-terminating and non-repeating
d)
Rational
| | Nidhi Bhatt answered |
Assume √2 is rational, so √2 = a/b with integers a,b in lowest terms and b ≠ 0.
Squaring gives 2 = a2/b2, hence a2 = 2b2. Thus a2 is even, so a is even; write a = 2k.
Then (2k)2 = 4k2 = 2b2, so b2 = 2k2, which implies b is even.
Both a and b being even contradicts that a/b is in lowest terms. Therefore √2 is irrational.
Therefore its decimal expansion is non-terminating and non-repeating.
How many Pythagorean triples can be generated from (3, 4, 5)?- a)Only one
- b)Finite
- c)Infinite
- d)None
Correct answer is option 'C'. Can you explain this answer?
a)
Only one
b)
Finite
c)
Infinite
d)
None
| | Nidhi Bhatt answered |
Multiply the primitive triple (3, 4, 5) by any positive integer k. This yields (3k, 4k, 5k), which satisfies (3k)2 + (4k)2 = (5k)2 for every k ≥ 1.
Therefore, infinitely many Pythagorean triples can be generated from (3, 4, 5) by scaling with k ∈ ℕ.
Which of the following numbers is rational?- a)√2
- b)√3
- c)√16
- d)√5
Correct answer is option 'C'. Can you explain this answer?
a)
√2
b)
√3
c)
√16
d)
√5
| | Nidhi Bhatt answered |
A rational number can be expressed as a fraction p/q where p and q are integers and q ≠ 0.
√16 = 4, and 4 = 4/1 is a rational number.
The square roots of integers that are not perfect squares (√2, √3, √5, ...) are irrational.
Therefore, √16 is rational; option C is correct.
Which of the following cannot form a right-angled triangle?- a)5, 12, 13
- b)8, 15, 17
- c)7, 24, 25
- d)6, 8, 15
Correct answer is option 'D'. Can you explain this answer?
a)
5, 12, 13
b)
8, 15, 17
c)
7, 24, 25
d)
6, 8, 15
| | Nidhi Bhatt answered |
To test, check whether the sum of squares of the two smaller sides equals the square of the largest side (Pythagoras).
Option A: 52 + 122 = 25 + 144 = 169 = 132, so it forms a right triangle.
Option B: 82 + 152 = 64 + 225 = 289 = 172, so it forms a right triangle.
Option C: 72 + 242 = 49 + 576 = 625 = 252, so it forms a right triangle.
Option D: largest side 15; 62 + 82 = 36 + 64 = 100 ≠ 225 = 152, so it does not form a right triangle.
Therefore, option D is correct.
If the area of a square becomes three times its original area, the new side is:- a)3a
- b)√3 a
- c)2a
- d)9a
Correct answer is option 'B'. Can you explain this answer?
a)
3a
b)
√3 a
c)
2a
d)
9a
| | Nidhi Bhatt answered |
Let the original side be a. The original area = a2.
The new area = 3a2.
The new side = √(3a2) = a√3.
Therefore the new side = a√3, so option B is correct.
If the hypotenuse is 13 cm and one side is 5 cm, the other side is:- a)10 cm
- b)11 cm
- c)12 cm
- d)9 cm
Correct answer is option 'C'. Can you explain this answer?
a)
10 cm
b)
11 cm
c)
12 cm
d)
9 cm
| | Nidhi Bhatt answered |
For a right triangle, h2 = a2 + b2, where h is the hypotenuse.
Let the unknown side be x. Then 132 = 52 + x2.
169 = 25 + x2 ⇒ x2 = 144.
x = √144 = 12 cm.
Which set of numbers satisfies a² + b² = c²?- a)(4, 5, 6)
- b)(10, 24, 26)
- c)(7, 8, 9)
- d)(2, 3, 4)
Correct answer is option 'B'. Can you explain this answer?
a)
(4, 5, 6)
b)
(10, 24, 26)
c)
(7, 8, 9)
d)
(2, 3, 4)
| | Nidhi Bhatt answered |
Check option B: 102 + 242 = 100 + 576 = 676, and 262 = 676.
For the other options: 42 + 52 = 41 ≠ 62 (36); 72 + 82 = 113 ≠ 92 (81); 22 + 32 = 13 ≠ 42 (16).
Therefore, (10, 24, 26) (option B) satisfies a² + b² = c².
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