Number Properties : Squares - Right Triangles

The question is from the topic number systems / trigonometry.

Question

'a' and 'b' are the lengths of the base and height of a right angled triangle whose hypotenuse is 'h'. If the values of 'a' and 'b' are positive integers, which of the following cannot be a value of the square of the hypotenuse?

1. 13
2. 23
3. 37
4. 41

Correct Answer - 23. Choice (2)

Explanatory Answer

Pythagoras Theorem : The square of the hypotenuse = h² = a² + b²

As the problem states that 'a' and 'b' are positive integers, the values of a² and b² will have to be perfect squares. Hence we need to find out that value amongst the four answer choices which cannot be expressed as the sum of two perfect squares.

Choice 1 is 13. 13 = 9 + 4 = 3² + 2². Therefore, Choice 1 is not the answer as it is a possible value of h²

Choice 2 is 23. 23 cannot be expressed as the sum two numbers, each of which in turn happen to be perfect squares. Therefore, Choice 2 is the answer.

Choice 3 is 37. 37 = 36 + 1 = 6² + 1². Therefore, Choice 3 is not the answer as it is a possible value of h²

Choice 4 is 41. 41 = 25 + 16 = 5² + 4². Therefore, Choice 4 is not the answer as it is a possible value of h²

Correct answer choice (2)

Level of difficulty : Moderate

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