Mathematics Questions?

Question

Triangle ABC has area equal to 90√3/4 and perimeter equal to 30. Also, one of its angles is equal to 60∘. What is the product of the sides of ABC?

Answer 1

Given that in the triangle ABC, <A = 60°

So its Area  = (1/2)*AB*AC*sin(A)

                      = (1/2)bcsin(60°)        [since sin(60) = √3/2]

                      = (1/2) (90) (√3/2)

                      = 90√3/4

Taking sin 60 =√3/2 and simplifying, bc = 90 ------ (1)

By properties of triangle,

                                  Radius r = Area/semi perimeter

                                                   = 90√3/(4*15)

                                                   = (3√3)/2
Also r = (s - a)*tan(A/2)

           = (15 - a)tan(30)

          = (3√3)/2

Solving a = 21/2 -------- (2)

Thus from (1) & (2),   Product of the sides = (a)*(bc)

                                                                           = (21/2)*90

                                                                           = 21 x 45

                                                                           = 945

Therfore product of the 3 sides = 945.