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

Question

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

Aaliyah Anderson · Accepted Answer

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

Angle between Sides # A and B# is # /_c= pi/3=180/3=60^0# Angle between Sides # 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]

Savannah Aguilar · Answer

undefined The largest possible area of the triangle can be found using the formula for the area of a triangle: $ A = \frac{1}{2} \times b \times h $, where $ b $ is the base of the triangle and $ h $ is the corresponding height. Given that one side of the triangle has a length of 5, this side can be considered as the base. The height of the triangle can be determined using trigonometric ratios.

Since we know two angles of the triangle ($ \frac{\pi}{3} $ and $ \frac{\pi}{6} $), we can find the third angle by subtracting the sum of the given angles from $ \pi $. Then, we can use trigonometric ratios to find the height of the triangle corresponding to the base.

With the base and height, we can calculate the area of the triangle.