Determine the perimeter. Determine the perimeter of triangle ABC if AB=6, B=pi/4, C=pi/6.
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For the triangle ABC AB = 6, B=pi/4, C=pi/6.
As the angles of a triangle have a sum of pi. A = 7*pi/12
Use the property sin A/a = sin B/b = sin C/c
c = 6, C = pi/6, B = pi/4 and A = 7*pi/12
=> sin (pi/6)/6 = sin (pi/4)/b = sin (7*pi/12)/a
=> b = 8.485
=> a = 11.59
The perimeter of the triangle is 6 + 8.48 + 11.59 = 26.07
To determine the perimeter of the triangle ABC, we'll have to determine all the lengths of it's sides.
A = pi - pi/4 - pi/6
A = (12pi - 3pi - 2pi)/12
A = 7pi/12
A = pi/2 + pi/12
sin pi/12 = sqrt[(1 - cos pi/6)/2]
sin pi/12 = sqrt(2 - sqrt3)/2
Form sine theorem and we'll have:
AB/sin C = AC/sin B
6/sin pi/6 = AC/sin pi/4
AC* 1/2 = 6*sqrt2/2
AC = 6sqrt2
AC/sin B = BC/sin A
6sqrt2/ sin pi/4 = BC/[sqrt(2 - sqrt3)/2]
BC*sqrt2/2 = {6sqrt2*[sqrt(2 - sqrt3)]}/2
BC = 6[sqrt(2 - sqrt3)]
The perimeter P is:
P = AB + AC + BC
P = 6 + 6sqrt2 + 6[sqrt(2 - sqrt3)]
We'll factorize by 6:
P = 6[1+sqrt2+sqrt(2 - sqrt3)] units
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