If one side of a triangle is 7 with its perimeter equal to 18, and area equal to $$\sqrt {108} ,$$  then the other two sides are:

Solution (By Examveda Team)

a = 7 cm
2S = a + b + c = 18
S = 9 cm
a + b + c = 18
7 + b + c = 18
b + c = 11
b = 11 - c
$$\eqalign{ & \Delta = \sqrt {{\text{S}}\left( {{\text{S}} - {\text{A}}} \right)\left( {{\text{S}} - {\text{B}}} \right)\left( {S - {\text{C}}} \right)} \cr & \Delta = \sqrt {9\left( {9 - 7} \right)\left( {9 - 11 - {\text{C}}} \right)\left( {9 - {\text{C}}} \right)} \cr & \Delta = \sqrt {9 \times 2\left( {{\text{C}} - 2} \right)\left( {9 - {\text{C}}} \right)} \cr & \sqrt {108} = \sqrt {18\left( {{\text{C}} - 2} \right)\left( {9 - {\text{C}}} \right)} \cr} $$
6 = 9C - C² - 18 + 2C
24 = C² + 11C
C² - 11C + 24 = 0
C² - 8C - 3C + 24 = 0
C(C - 8) - 3(C - 8) = 0
(C - 3)(C - 8) = 0
C = 3, 8
B = 8, 3
Side = 3, 8