A triangle has two corners with angles of # (2 pi ) / 3 # and # ( pi )/ 4 #. If one side of the triangle has a length of #2 #, what is the largest possible area of the triangle?

Question

A triangle has two corners with angles of #  (2 pi ) /  3  # and # ( pi  )/  4   #.  If one side of the triangle has a length of #2  #, what is the largest possible area of the triangle?

Caroline Aucoin · Accepted Answer

Area of the largest possible triangle is #15.08# sq.unit.

Angle between Sides # A and B# is # /_c= (2pi)/3=120^0# Angle between Sides # B and C# is # /_a= pi/4=45^0 :.# Angle between Sides # C and A# is # /_b= 180-(120+45)=15^0# For largest area of triangle #2# should be smallest side , which is opposite to the smallest angle #(/_b=15^0)#, i.e #B=2# 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 ; B=2 :. A/sina=B/sinb# or #A/sin45=2/sin15 :. A= 2* sin45/sin15~~ 5.46(2dp)# Now we know sides #A=5.46 , B=2.0# and their included angle #/_c = 120^0#. Area of the triangle is #A_t=(A*B*sinc)/2# #:.A_t=(5.46*cancel2*sin 120)/cancel2 ~~ 4.73# sq.unit. Area of the largest possible triangle is #15.08# sq.unit [Ans]

Aaliyah Anderson · Answer

undefined To find the largest possible area of the triangle given that it has angles of $ \frac{2\pi}{3} $ and $ \frac{\pi}{4} $ and one side of length 2, you can use the formula for the area of a triangle:

$$ A = \frac{1}{2}ab\sin(C) $$

where $ a $ and $ b $ are the lengths of two sides of the triangle and $ C $ is the angle between them.

Given that one side has a length of 2, let's call it $ a $, and let $ b $ be another side. We're looking for the largest possible area, so we want to maximize $ b $.

1. **Find the Third Angle**:
   The sum of angles in a triangle is $ \pi $ radians. So, to find the third angle, subtract the given angles from $ \pi $.

$$ \text{Third angle} = \pi - \frac{2\pi}{3} - \frac{\pi}{4} $$

2. **Use Law of Sines**:
   Use the Law of Sines to find the ratio of sides $ a $ and $ b $.

3. **Maximize $ b $**:
   The side $ b $ should be maximized to get the largest area. It happens when $ b $ is opposite to the largest angle.

4. **Calculate Area**:
   Plug the values of $ a $, $ b $, and $ C $ into the area formula.

By following these steps, you can find the largest possible area of the triangle.