Question 340303: What is the perimeter of a triangle who's angles are 120, 30, and 30 degrees and the base length is 48√3m? Both angles of 30 degrees are to the left and right of the base.
Found 2 solutions by Theo, mananth:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! this is an isosceles triangle.
the base measures 48*sqrt(3) meters.
the angles are 30, 30, and 120.
the angle opposite the base is the 120 degree angle.
you can use the law of sines or you can divide the isosceles triangle into 2 right triangles by dropping a perpendicular from the angle opposite the base.
if you drop the perpendicular, then you get 2 right triangles, each of which has one 60 degree angle, one 30 degree angle and one 90 degree angle.
in a 30/60/90 degree triangle, the sides measure as follows in relationship to each other:
sine 30 = 1/2 = opposite / hypotenuse
cosine 30 = sqrt(3)/2 = adjacent / hypotenuse
you have the base of the isosceles triangle is equal to 48*sqrt(3).
take half of that to get 24*sqrt(3)
24*sqrt(3) becomes the measure of the adjacent side to each 30 degree angle of each right triangle.
since cosine 30 = sqrt(3)/2 = adjacent / hypotenuse, and adjacent = 24 * sqrt(3), then you get:
cosine 30 = sqrt(3)/2 = 24*sqrt(3)/hypotenuse
divide both sides of this equation by sqrt(3) and you get:
1/2 = 24/hypotenuse
multiply both sides of this equation by 2 to get:
1 = 48 / hypotenuse
multiply both sides of this equation by hypotenuse to get:
hypotenuse = 48
since the hypotenuse is one of the equal legs of the isosceles triangle, then you get a perimeter for the isosceles triangle of 48 + 48 + 48 * sqrt(3) which is equal to 96 + 48*sqrt(3)
if you use the law of sines, you should get the same answer.
the law of sines states that a/sin(a) = b/sin(b) = c/sin(c)
assume side a is opposite the 120 degree angle.
your formula becomes:
48*sqrt(3)/sin(120) = b/sin(30) = c/sin(30)
since 48*sqrt(3)/sin(120) = 96, then your formula becomes:
96 = b/sin(30) = c/sin(30)
if 96 = b/sin(30), this means that b = 96*sin(30).
since sin(30) = 1/3, this means that b = 48.
this also means that c = 48, and your final formula becomes:
96 = 48/sin(30) = 48/sin(30).
add all 3 sides together to get the perimeter of 96 + 48*sqrt(3)
since you did it both ways and got the same answer each time, then the answer is probably good.
go back to your right triangles.
the hypotenuse of each right triangle is 48.
take the cosine of 30 degrees to get 1/2 the base of the isosceles triangle.
cosine (60) = adjacent / hypotenuse = adjacent / 48.
multiply both sides of this equation by 48 to get:
adjacent = 48 * cos(60).
since cos(60) = .5 = 1/2, this means that adjacent = 24*sqrt(3).
since adjacent is 1/2 the base, this means the base of the isosceles triangle is 48*sqrt(3).
all ways point to the perimeter of the triangle being 48 + 48 + 48*sqrt(3) = 96 + 48*sqrt(3).
Answer by mananth(16946)  (Show Source):
You can put this solution on YOUR website! Let ABC be isoscles triangle.
AB = AC
BC = base = 48sqrt3
..
A perpendicular drawn from angle A bisects the base. at M
..
So AMC is a right triangle.
MC = 1/2 base = 24 sqrt3
..
Angle C = 30 deg.
MC/CA = cos 30
CA = MC / cos 30
CA = 24sqrt3/cos 30
CA = 48
AB = 48
BC = 48sqrt3
Perimeter = 48(2+sqrt3)
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