Two corners of a triangle have angles of #pi / 8 # and # pi / 12 #. If one side of the triangle has a length of #9 #, what is the longest possible perimeter of the triangle?

Question

Two corners of a triangle have angles of #pi  /  8  # and #  pi   /  12  #.  If one side of the triangle has a length of #9  #, what is the longest possible perimeter of the triangle?

Aurora Ayala · Accepted Answer

Longest possible perimeter is #43.48# unit.

Angle between Sides # A and B# is # /_c= pi/8=180/8=22.5^0# Angle between Sides # B and C# is # /_a= pi/12=180/12=15^0 :.# Angle between Sides # C and A# is # /_b= 180-(22.5+15)=142.5^0# For largest perimeter of triangle #9# should be smallest side , which is opposite to smallest angle #:.A=9# unit. 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=9 :. A/sina=B/sinb# or #9/sin 15=B/sin142.5 or B= 9* (sin142.5/sin15) ~~ 21.17 (2dp) # Similarly #A/sina=C/sinc # or #9/sin15=C/sin22.55 or C= 9* (sin22.5/sin15) ~~ 13.31 (2dp) # Perimeter #P=A+B+C =9+21.17+13.31 ~~43.48# unit. Longest possible perimeter is #43.48# unit. [Ans]

Savannah Aguilar · Answer

undefined To find the longest possible perimeter of the triangle, you need to calculate the lengths of the other two sides using trigonometric ratios. Since you have the angles of the triangle, you can use trigonometric functions to find the lengths of the other sides.

First, let's find the third angle of the triangle using the fact that the sum of the angles in a triangle is always π radians (180 degrees). So, the third angle can be found by subtracting the given angles from π radians.

Third angle = π - (π/8 + π/12)

Now, you can use trigonometric ratios to find the lengths of the other two sides. Let's call the side opposite the angle π/8 as 'x' and the side opposite the angle π/12 as 'y'.

For angle π/8:
x/sin(π/8) = 9
x = 9 * sin(π/8)

For angle π/12:
y/sin(π/12) = 9
y = 9 * sin(π/12)

Now, you have the lengths of the other two sides. Calculate their values, and then find the perimeter by adding all three side lengths together. Finally, choose the maximum perimeter among the possibilities to find the longest possible perimeter of the triangle.