Question

Two corners of a triangle have angles of `( pi )/ 3` and `( pi ) / 4`. If one side of the triangle has a length of `9`, what is the longest possible perimeter of the triangle?

Answer 1

Perimeter is `32.314`

Explanation:

As two angles of a triangle are `pi/3` and `pi/4`, the third angle is

`pi-pi/3-pi/4=(12-4-3)pi/12=(5pi)/12`

Now for the longest possible perimeter, the given side say `BC`, should be the smallest angle `pi/4`, let this be `/_A`. Now using sine formula

`9/sin(pi/4)=(AB)/sin(pi/3)=(AC)/sin((5pi)/12)`

Hence `AB=9xxsin(pi/3)/sin(pi/4)=9xx(sqrt3/2)/(sqrt2/2)=9xx1.732/1.414=11.02`

and `AC=9xxsin((5pi)/12)/sin(pi/4)=9xx0.9659/(1.4142/2)=12.294`

Hence, perimeter is `9+11.02+12.294=32.314`