problem for topper

Geometry Level 2

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?

37 23 13 41

3 solutions

. .
May 9, 2021

$37 = 1 ^ { 2 } + 6 ^ { 2 }$

$13 = 2 ^ { 2 } + 3 ^ { 2 }$

$41 = 4 ^ { 2 } + 5 ^ { 2 }$

Hence, $23$ cannot be.

Sanyam Garg
Jun 5, 2020

[paragraph 1] There is a simple way that you directly think about the possible triplets..........but i am going to tell you guys awesome property of pythagoras -that is , [paragraph 2] ( 2m, m^2-1,m^2+1) .now you can directly start putting value of m=1,2,3,4,5,6......you will yourself got the answer .i.e 23

Shivam Bhatia
Dec 26, 2014

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.

In simple words, we have $(5,12,13),(12,35,37),(9,40,41)$ as pythagorean triplets but $(a,b,23)$ can't be a pythagorean triplet $\forall a,b\in \mathbb{Z^+}$ , so the correct answer is $\boxed{23}$

The only way that I know to solve this type of problem is to verify each option using a calculator!

Prasun Biswas - 6 years, 5 months ago

How $13 = 9 + 4 = 32 + 22$ ?

Maybe you have to use exponent ?

. . - 1 month ago

problem for topper

3 solutions

0 pending reports