Question

In triangle PQR, \[\angle P:\angle Q:\angle R = 5:6:7\], without finding the angles of the triangle, name its largest side.
A. PQ
B. QR
C. PR
D. None of these

Answer 1

Hint: - According to the property of triangles “Side opposite to longest angle of the triangle is longest side of the triangle” and side opposite to the shortest angle of the triangle is the shortest side of the triangle.

Complete step by step solution:
As we know that the ratio is the comparison of some values, it shows us that when we have this much something then we will need to have that much something.
Like if the ratio of A and B is A : B = x : y, then it means if A = x then B = y. So, if x is greater than y, then the original value of A must be greater than the original value of B and if x is less than y then original value of A must be less than the original value of B.
So, we are given the ratio of angle P, Q and R is 5 : 6 : 7. It means that \[\angle P\] is smallest and \[\angle R\] is largest.
Now as we know that side opposite to the longest angle is the largest side of the triangle.
So, let us draw the triangle PQR, so that we can find the side opposite to the \[\angle R\].

All the angles in the above triangle are denoted in ratio. So, as we can see from the above triangle that side PQ is opposite to \[\angle R\]. So, side PQ must be the largest side.
Hence, the correct option will be A.
Note: - Whenever we come up with this type of problem and if we are given the ratio of angles of the triangle and asked to find the smallest or largest side then smallest side will be the side opposite to the angle whose ratio is smallest and largest side will be the side opposite to the angle whose ratio is largest of all. So, this will be the easiest and efficient way to find the solution of the problem.