A triangle has two corners with angles of # ( pi ) / 2 # and # ( 5 pi )/ 12 #. If one side of the triangle has a length of #9 #, what is the largest possible area of the triangle?

Question

A triangle has two corners with angles of #  ( pi ) /  2  # and # ( 5 pi  )/  12   #.  If one side of the triangle has a length of #9  #, what is the largest possible area of the triangle?

Aurora Ayala · Accepted Answer

The area is #=151.1u^2#

The third angle of the triangle is

#=1/2pi-5/12pi=1/12pi#

To have the largest possible area, the side of length #9# is opposite the smallest angle, i.e, #1/12pi#

Let the side opposite the angle #5/12pi# be #=a#

Applying the sine rule to the triangle, #a/sin(5/12pi)=9/sin(1/12pi)#

Therefore,

#a=9*sin(5/12pi)/sin(1/12pi)=33.6#

The area of the triangle is

#A=1/2*9*33.6=151.1#

Aurora Ayala · Answer

undefined To find the largest possible area of the triangle, use the formula for the area of a triangle: $ A = \frac{1}{2} \times b \times h $, where $ b $ is the base and $ h $ is the height.

In this case, one side of the triangle has a length of 9, which can be considered as the base. To find the height, you need to determine the length of the altitude from the vertex opposite the base to the base itself.

To find the height, use trigonometry. The height can be calculated using the formula $ h = \text{side} \times \sin(\text{angle}) $, where the angle is the angle opposite to the side whose length is known.

For the given triangle, the angle opposite to the side with length 9 is $ \frac{5\pi}{12} $. Thus, $ h = 9 \times \sin\left(\frac{5\pi}{12}\right) $.

Calculate the value of $ \sin\left(\frac{5\pi}{12}\right) $, then multiply it by 9 to find the height.

Finally, use the formula for the area of the triangle to calculate the largest possible area.