Question Number 113847 by deepraj123 last updated on 15/Sep/20

The two adjacent sides of a cyclic  quadrilateral are 2 and 5 and the  angle between them is 60°. If the third  side is 3, the remaining fourth side is

Answered by 1549442205PVT last updated on 16/Sep/20

Commented by1549442205PVT last updated on 16/Sep/20

i)Case CD=3  From the hypothesis   AB=2,BC=5,ABC^(�) =60°,CD=3 and  cosine theorem we have  AC^2 =AB^2 +BC^2 −2AB.BC.cos60°  ⇔AC^2 =2^2 +5^2 −2.2.5.(1/2)=19.On the  other hand,ADC^(�) =180°−ABC^(�) =120°  (since quadrilateral ABCD inscribed)  Apply cosine theorem for ΔACD with  AC=(√(19)),AD=3,ADC^(�) =120°we get  AC^2 =CD^2 +AD^2 −2AD.CDcos120°  ⇔19=3^2 +AD^2 −2.3.AD.(−(1/2))  ⇔AD^2 +3AD−10=0.Solve this equation  Δ=9+4.10=7^2 ⇒AD=(−3+7)/2=2  Thus,fourth side equal to 2(cm)  ii)Case AD=3 we do similarly also  getting CD=2