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:
Answer & explanation
Correct answer: option 4
One side of triangle = 7 cm
Perimeter of triangle = 18 cm
Area of triangle = $\sqrt{108}$
And we know that,
Area of a triangle = \(\sqrt {s(s-a)(s-b)(s-c) }\)
according to the question,
18 = (7 + b + c)
= (b + c) = 11
= c = 11 - b
S = \(\frac{18}{2}\) = 9
\(\sqrt {108}\) = \(\sqrt {[(9)(9 - 7)(9 - b)(9 - c)] }\)
Squaring both sides,
108 = 18 × (9 - b) × (9 - c)
Put the value of c in above equation,
6 = (9 - b)[9 - (11 - b)]
= 6 = (9 - b)(b - 2)
= 6 = 9b + 2b - b2 - 18
= b2 - 11b + 24 = 0
= b2 - 8b - 3b + 24 = 0
= b(b - 8) - 3(b - 8) = 0
= (b - 8)(b - 3) = 0
= b = 8, 3
Take side b = 8
Then, c = 11 - 8 = 3
So, the other two sides are = 3 and 8