Question

Which of the following can be the sides of a right angled triangle?
A) \[35,17,18\]
B) $39,19,18$
C) $35,27,18$
D) $41,40,9$

Answer 1

Hint: The lengths of the sides of a right triangle are related by Pythagoras theorem. So we can check the options using the theorem. Thus we can find the right option.

Formula used: Pythagoras theorem:
For a right angled triangle, $bas{e^2} + altitud{e^2} = hypotenus{e^2}$
where hypotenuse is the side opposite to $90^\circ $ angle.

Complete step-by-step answer:
We are given four sets of angles.
We have to find which set represents sides of a right angled triangle.
The lengths of the sides of a right triangle are related by Pythagoras theorem.
The theorem says that, for a right angled triangle, $bas{e^2} + altitud{e^2} = hypotenus{e^2}$
where hypotenuse is the side opposite to $90^\circ $ angle.
So we can check each set one by one.
A) \[35,17,18\]
Here the larger number is $35$. So we can check whether ${17^2} + {18^2} = {35^2}$.
${17^2} + {18^2} = 289 + 324 = 613$
But ${35^2} = 1225$
So the theorem does not hold.
Therefore, this cannot be the sides of a right angled triangle.
B) $39,19,18$
Here the larger number is $39$. So we can check whether ${19^2} + {18^2} = {39^2}$.
${19^2} + {18^2} = 361 + 324 = 685$
But ${39^2} = 1521$
So the theorem does not hold.
Therefore, this cannot be the sides of a right angled triangle.
C) $35,27,18$
Here the larger number is $35$. So we can check whether ${27^2} + {18^2} = {35^2}$.
${27^2} + {18^2} = 729 + 324 = 1053$
But ${35^2} = 1225$
So the theorem does not hold.
Therefore, this cannot be the sides of a right angled triangle.
D) $41,40,9$
Here the larger number is $41$. So we can check whether ${40^2} + {9^2} = {41^2}$.
${40^2} + {9^2} = 1600 + 81 = 1681$
Also ${41^2} = 1681$
So the theorem holds.
Therefore, this can be the sides of a right angled triangle.

$\therefore $ Option D is the correct answer.

Note: This is the method to find the correct option. But for objective type exams, that is, if the step is not necessary, we can find the answer in a simpler way. Instead of adding each square, just add the unit digits first. If the sum of the unit digits is not satisfying that option is wrong.
For example,
For \[35,17,18\], ${17^2}$ ends with $9$ and ${18^2}$ ends with $4$. So their sum ends with $3$, since $9 + 4 = 13$. But ${35^2}$ ends with $5$. So this cannot be right. We cannot say an option is right only using this. But we can eliminate the wrong options.