Solution-
As given in △ABC,
[tex]m\angle A>m\angle B>m\angle C[/tex]
As from the properties of trigonometry we know that, the greater the angle is, the greater is the value of its sine. i.e
[tex]\sin A>\sin B>\sin C[/tex]
According to the sine law,
[tex]\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}[/tex]
In order to make the ratio same, even though m∠A>m∠B>m∠C, a must be greater than b and b must be greater than c.
[tex]\Rightarrow a>b>c[/tex]
Also given that its perimeter is 30. Now we have to find out whose side length is 7. So we have 3 cases.
Case-1. Length of a is 7
As a must be the greatest, so b and c must be less than 7. Which leads to a condition where its perimeter won't be 30. As no 3 numbers less than 7 can add up to 30.
Case-2. Length of b is 7
As b is greater than c, so c must 6 or less than 6. But in this case the formation of triangle is impossible. Because the triangle inequality theorem states that the sum of any 2 sides of a triangle must be greater than the measure of the third side. If b is 7 and c is 6, then a must be 17. So no 2 numbers below 7 can add up to 17.
Case-3. Length of c is 7
As this is the last case, this must be true.
Therefore, by taking the aid of process of elimination, we can deduce that side c may have length 7.
