Question 173124: This is the question:
If one acute angle of a right triangle is twice as big as the other, then the one leg of the triangle is twice the other. (We have to answer Always, Sometimes, or never. If it is Sometimes or Never then we have to change it to make it an Always statement).
I know that it is Never but I can't find a triangle that makes it an Always statement. (We have to use GSP for this).
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! If one acute angle of a right triangle is twice as big as the other, then the one leg of the triangle is twice the other. (We have to answer Always, Sometimes, or never. If it is Sometimes or Never then we have to change it to make it an Always statement).
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If one acute angle is twice the other, one is 30 degrees; the other is 60 degrees.
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Then by the Law of Sines:
smaller/sin(30) = larger/sin(60)
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Assume smaller is x and larger is 2x.
Then x/sin(30) = (2x)/sin(60)
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Then x/(1/2) = (2x)/(sqrt(3/2)
Then 2x = 2x/(sqrt(2/3)
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This can only be true if x = 0
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But x cannot be zero if it is an acute angle.
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Therefore the assumption is wrong.
The sides opposite two complementary angles that are 2:1 cannot be 2:1.
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Cheers,
Stan H.
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