SOLUTION: if one side of a triangle is double the other and the angles opposite these sides differ by 60. show that the triangle is right angled

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Question 261230: if one side of a triangle is double the other and the angles opposite these sides differ by 60. show that the triangle is right angled
Answer by drk(1908) About Me  (Show Source):
You can put this solution on YOUR website!
Let A, B, C be the sides and y, y-60 be the angles.
we know
(1) side A = x, side C = 2x
(2) angle a = y-60, angle c = y
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using the fact that all angles add to 180, we know that 180 - y - (y-60) = angle b, which is
angle b = 240 - 2y.
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using law of sines we get
(i)%282x%2Fsiny%29+=+%28x%2Fsin%28y-60%29%29
cross multiply to get
(ii) 2xsin%28y-60%29+=+xsin%28y%29
use the sin subtraction identity to get
(iii)2x%2A%28sin%28y%29cos%2860%29+-+cos%28y%29+sin%2860%29%29+=+xsin%28y%29
simplify to get
(iv) xsin%28y%29+-+%28sqrt%283%29%2F2%29cos%28y%29+=+xsin%28y%29
subtract to get
(v) -+%28sqrt%283%29%2F2%29cos%28y%29+=+0
this means that cos(y) = 0
and y = 90 or 180.
SInce y cannot be 180 for a triangle, we have
y = 90.
This tells us that the triangle MUST be right.