Two corners of a triangle have angles of $(5 pi )/ 12$ and $pi / 6$ . If one side of the triangle has a length of 3, what is the longest possible perimeter of the triangle?

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Answer 1 · 2018-02-05T11:51:24

Perimeter of the longest possible triangle is $14.6$ unit.

Angle between Sides $A and B$ is #

$/_c= (5pi)/12=(5*180)/12=75^0$

Angle between Sides $B and C$ is $/_a= pi/6=180/6=30^0 :.$

Angle between Sides $C and A$ is

$/_b= 180-(75+30)=75^0$ . For largest perimeter of

triangle $3$ should be smallest side , which is opposite

to the smallest angle $/_a=30^0:.A=3$ . 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, $A/sina = B/sinb=C/sinc$

$:. A/sina=B/sinb or 3/sin30 = B/sin 75: B = (3*sin75)/sin30$ or

$B~~5.80 ; B/sinb=C/sinc or 5.80/sin75=C/sin75$

$:. C~~ 5.8 :. A=3.0 , B~~ 5.8 , C~~ 5.8$ . Perimeter of the

triangle is $P_t=A+B+C ~~ 3.0+5.8+5.8=14.6$ unit.

Perimeter of the longest possible triangle is $14.6$ unit [Ans]

Two corners of a triangle have angles of (5 pi )/ 12 (5 pi )/ 12 and pi / 6 pi / 6 . If one side of the triangle has a length of 3, what is the longest possible perimeter of the triangle?