# TrianglesDetermine 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**

We need to know all the lengths of the sides of triangle, in order to get the value of it's perimeter.

We'll begin by computing the angle A.

Since the sum of the angles of a triangle is 180 degrees = pi radians, we'll get A:

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

We'll apply sine theorem and we'll get:

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**