Question 261230
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){{{(2x/siny) = (x/sin(y-60))}}}
cross multiply to get
(ii) {{{2xsin(y-60) = xsin(y)}}}
use  the sin subtraction identity to get
(iii){{{2x*(sin(y)cos(60) - cos(y) sin(60)) = xsin(y)}}}
simplify to get
(iv) {{{xsin(y) - (sqrt(3)/2)cos(y) = xsin(y)}}}
subtract to get
(v) {{{- (sqrt(3)/2)cos(y) = 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.