Question

If the ratio of three sides of a triangle is $\sqrt{3}:\sqrt{4}:\sqrt{5},$ then find whether it is a right angle triangle or not.

Answer 1

Let the sides be $\sqrt{3}x\sqrt{4}x$ and $\sqrt{5}x.$
Let us assume that the sides form a right-angled triangle.
But, from Pythagoras' theorem, we know that, the square of the hypotenuse(longest side) is equal to the sum of the squares of the other two sides in a right-angled triangle.
Then, we must have $(\sqrt{3}x{)}^2+(\sqrt{4}x{)}^2=(\sqrt{5}x{)}^2.$
$\Rightarrow 3x^2+4x^2=5x^2$
$\Rightarrow 2x^2=0$
$\Rightarrow x=0,$ not true.
We have arrived at this conclusion due to our wrong assumption.
Thus, the given sides do not form a right-angled triangle.