Question

Two corners of a triangle have angles of `pi / 8` and `pi / 12`. If one side of the triangle has a length of `9`, what is the longest possible perimeter of the triangle?

Answer 1

Longest possible perimeter is `43.48` unit.

Explanation:

Angle between Sides `A and B` is `/_c= pi/8=180/8=22.5^0`

Angle between Sides `B and C` is `/_a= pi/12=180/12=15^0 :.`

Angle between Sides `C and A` is

`/_b= 180-(22.5+15)=142.5^0` For largest perimeter of

triangle `9` should be smallest side , which is opposite to

smallest angle `:.A=9` unit.

The sine rule states if `A, B and C` are the lengths of the sides

and opposite angles are `a, b and c` in a triangle, then:

`A/sinA = B/sinb=C/sinc ; A=9 :. A/sina=B/sinb` or

`9/sin 15=B/sin142.5 or B= 9* (sin142.5/sin15) ~~ 21.17 (2dp)`

Similarly `A/sina=C/sinc` or

`9/sin15=C/sin22.55 or C= 9* (sin22.5/sin15) ~~ 13.31 (2dp)`

Perimeter `P=A+B+C =9+21.17+13.31 ~~43.48` unit.

Longest possible perimeter is `43.48` unit. [Ans]