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

James Atkins · Accepted Answer

#frac(81)(4)# To create a triangle with maximum area, we want the side of length #9# to be opposite of the smallest angle. Let us denote a triangle #DeltaABC# with #angleA# having the smallest angle, #frac(pi)(12)#, or #15# degrees. Thus, #BC=9#, and since the triangle is isoceles, #AC=BC=9#. Angle #C# measures #150# degrees (#180-15-15#). Now, we may find the area of the triangle through the formula #Area=frac(1)(2)ab sin(C)#, or #frac(1)(2)AC*BC sin(C)=frac(81)(4)#.

Sophia Alfred · Answer

#81/4=20.25#

Given that two of the angles are #pi/12#, the triangle is isosceles, and the third angle is #pi-pi/12-pi/12=(5pi)/6#

If one side of the triangle has a length of #9# unit, then its maximum area will occur when this length is opposite to one of its smaller angles, as shown in the diagram.

Area of a triangle #A=1/2ab*sinx#,
where #a and b# are two sides of the triangle and #x# is the included angle between the two sides #a and b#.

Here, given the triangle is isosceles, #=> a=b=9#.
#=> A=1/2*a^2*sinx#

#=># Area of the triangle #A= 1/2*9^2*sin((5pi)/6)=1/2*81*1/2=81/4=20.25#

Evelyn Arboleda · Answer

undefined To find the largest possible area of the triangle, we can use the formula for the area of a triangle: A = 1/2 * base * height.

Given that two angles of the triangle are π/12 and π/12, we know that the third angle must be π - (π/12) - (π/12) = π - π/6 = 5π/6.

Now, we can use the Law of Sines to find the lengths of the other two sides of the triangle. Let x be the length of one of these sides. We have:

sin(π/12) / 9 = sin(5π/6) / x

Solving for x, we get:

x = 9 * (sin(5π/6) / sin(π/12))

Now, we can use the formula for the area of a triangle to find the largest possible area. The base of the triangle is 9, and the height can be found using trigonometry:

height = x * sin(π/12)

Substituting the values of x and base into the formula for the area, we get:

A = 1/2 * 9 * (9 * (sin(5π/6) / sin(π/12))) * sin(π/12)

Calculating this expression will give us the largest possible area of the triangle.