Question

Out of the following which is a Pythagorean triplet ?

• A. (1, 5, 10)
• B. (3, 4, 5)
• C. (2, 2, 2)
• D. (5, 5, 2)

Hint:

Memorizing common Pythagorean triplets like (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25) can significantly speed up problem-solving in geometry and trigonometry sections of competitive exams.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A Pythagorean triplet consists of three positive integers \(a\), \(b\), and \(c\) such that the sum of the squares of the two smaller integers equals the square of the largest integer.

Step 2: Key Formula or Approach:
The formula to verify a Pythagorean triplet is \(a^2 + b^2 = c^2\), where \(c\) is the greatest of the three numbers. We will test each option against this formula.

Step 3: Detailed Explanation:
Let's check each option:
(A) (1, 5, 10):
Here, \(a = 1\), \(b = 5\), and \(c = 10\).
\[ 1^2 + 5^2 = 1 + 25 = 26 \] \[ 10^2 = 100 \] Since \(26 \neq 100\), this is not a Pythagorean triplet.
(B) (3, 4, 5):
Here, \(a = 3\), \(b = 4\), and \(c = 5\).
\[ 3^2 + 4^2 = 9 + 16 = 25 \] \[ 5^2 = 25 \] Since \(25 = 25\), this is a Pythagorean triplet.
(C) (2, 2, 2):
All numbers are equal. This represents the sides of an equilateral triangle, not a right-angled triangle. Also, \(2^2 + 2^2 = 8\), which is not equal to \(2^2 = 4\).
(D) (5, 5, 2):
The two smaller numbers are 2 and 5. The largest is 5.
\[ 2^2 + 5^2 = 4 + 25 = 29 \] \[ 5^2 = 25 \] Since \(29 \neq 25\), this is not a Pythagorean triplet.

Step 4: Final Answer:
Based on the calculations, the only set of numbers that satisfies the Pythagorean theorem is (3, 4, 5).