# 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?

- 13
- 23
- 37
- 41

Correct Answer -

**23**. Choice (2)

#### Explanatory Answer

Pythagoras Theorem : The square of the hypotenuse = h

^{2} = a

^{2} + b

^{2}
As the problem states that 'a' and 'b' are positive integers, the values of a

^{2} and b

^{2} 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 = 3

^{2} + 2

^{2}. Therefore, Choice 1 is not the answer as it is a possible value of h

^{2}
**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 = 6

^{2} + 1

^{2}. Therefore, Choice 3 is not the answer as it is a possible value of h

^{2}
**Choice 4 is 41. **41 = 25 + 16 = 5

^{2} + 4

^{2}. Therefore, Choice 4 is not the answer as it is a possible value of h

^{2}
Correct answer choice (2)

Level of difficulty : Moderate

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