A triangle has two corners with angles of $\frac{π}{3}$ $( pi ) / 3$ and $\frac{π}{6}$ $( pi )/ 6$ . If one side of the triangle has a length of $5$ $5$ , what is the largest possible area of the triangle?

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Answer 1 · 2018-01-19T12:08:33

Area of the largest possible triangle is $21.65$ $21.65$ sq.unit.

Angle between Sides $A and B$ $A and B$ is $∠ c = \frac{π}{3} = \frac{180}{3} = 60^{0}$ $/_c= pi/3=180/3=60^0$

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

Angle between Sides $C and A$ is $/_b= 180-(60+30)=90^0$

For largest area of triangle $5$ should be smallest size. which

is opposite to the smallest angle , $:. A=5$ . 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,

$A/sina = B/sinb=C/sinc ; A/sina=C/sinc$ or

$5/sin30=C/sin60 :. C= 5* sin60/sin30~~ 8.66(2dp)$

Now we know sides $A=5 , C=8.66$ and their included angle

$/_b = 90^0$ . Area of the triangle is $A_t=(A*C*sinb)/2$

$:.A_t=(5*8.66*sin90)/2 ~~ 21.65$ sq.unit.

Area of the largest possible triangle is $21.65$ sq.unit [Ans]

A triangle has two corners with angles of ( pi ) / 3 π3 ( pi ) / 3 and ( pi )/ 6 π6 ( pi )/ 6 . If one side of the triangle has a length of 5 55 , what is the largest possible area of the triangle?