In a triangle, the lengths of the two larger sides are 10 and 9, respectively. If the angles are in A.P, then the length of the third side can be (a) `5-sqrt(6)` (b) `3sqrt(3)` (c) `5` (d) `5+sqrt(6)`

Options

A. `5pmsqrt(6)`
B. `3sqrt(3)`
C. 5
D. none of these

Answer

A

Explanation

Let a = 10 and b = 9
It is given that the angles are in A.P.
`therefore 2B=A+C`
implies 3B=A+B+C
`implies3B=180^(@) " "[because A+B+C=180^(@)]`
`implies B= 60^(@)`
`implies cosB=(1)/(2)`
`implies (a^(2)+c^(2)-b^(2))/(2ac)=(1)/(2)`
`implies a^(2)+c^(2)-b^(2)=ac`
`implies 100+c^(2)-81=10cimpliesc^(2)-10c+19=0impliesc=5pm sqrt(6)`