Is triangle ABC a right triangle?

(1) The ratio of the largest to the smallest side of the triangle is 5 : 3.
(2) The ratio of the smallest side to the next larger side is 3 : 4.
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Question stem:- Is ABC a right triangle?

St1:- The ratio of the largest to the smallest side of the triangle is 5 : 3.
Note:- Ratio relationship with the third side is not given.
Length of third side lies between 3x to 5x (both 3x and 5x are exclusive)

a)Say x=1,then measure of third side=4 , so it's a right angled triangle(3,4,5)
b)Say x=1,when measure of third side=Any value between 3 to 5 (not necessarily integers, say 3.5,4.1 etc), then it's not a right angled triangle Insufficient.

St2:- The ratio of the smallest side to the next larger side is 3 : 4
As per triangle inequality theorem, the measure of third side can be any number less than 7x(=3x+4x) and greater than x(=4x-3x).

say x=1, then the sides of the triangle would be 3,4, and 1<3rd side<7.
Insufficient.

Combined,
Sides of the triangle are in the ration 3k:4k:5k.

So, the triangle ABC is a right angled triangle.

Ans. (C)
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