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Prove that if the segment in front of $30$ degree is $\frac{1}{2}$ another segment the triangle is a right triangle.

In the book it is written that the median is the key to solve these kind of questions but I can't solve it using that any hints?

enter image description here

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    Maybe remember the [law of sines](https://en.wikipedia.org/wiki/Law_of_sines) to find the angle opposite to the $2x$ side. – 2017-01-26
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    Construct a congruent triangle reflected across the long leg. Prove that the two triangles together form and equilateral triangle. And the median is perpendicular to the base. – 2017-01-26
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    consider an equilateral triangle, draw a line perpendicular line from one of the sides to opposite vertex then you will get it – 2017-01-26
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    @DougM By long leg you mean $2x$?But then it won't be equilateral? – 2017-01-26
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    @dxiv we didn't learn laws of $\sin$ yet.But we have learned laws of $\cos$. – 2017-01-26
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    @TahaAkbari 2x would be the hypotenuse (assuming that it is indeed an equlateral triangle. When I said long leg, I intended the one that was otherwise unidentified. – 2017-01-26

3 Answers 3

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PICTURE ONE:

enter image description here

$\angle CAB=30°$ and $AB=2BC$

Assuming $\angle ACB \ne 90°$. We find a point $D$ lying on ray $AC$ such that $\angle BDA=90°$.

Because $\angle CAB=30°$ and $\angle BDA=90°$, $AB=2BD$

However, we have been told that $AB=2BC$.

As a result, $BD=BC$. But $BD$ is perpendicular to $AC$......

CONTRADICTION!

PICTURE TWO:

enter image description here

$\angle BDA$ is assumed to be $90°$, while $\angle CAB=30°$, so we have $\angle DAB+\angle BDA=240°$.

CONTRADICTION!

Now the proposition is proved.

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    Oh, I made such a captial mistake! I'll try to find anthor solution. – 2017-01-26
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    I edited to fix the proof. – 2017-01-26
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Let the unknown angle be R. By Sine Rule

$$ \dfrac{\sin R}{2x}= \dfrac{\sin 30^0}{x} \rightarrow \sin R = 1, R= 90^0. $$

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enter image description here

You suppose to prove x=2y, extend your triangle to form another congruent triangle(the brown coloured triangle). Now angle a=60 since sum of the angle of the triangle(black one) should be 180. Since left side triangle is nothing but a mirror image of black triangle, hence b=60 and c=30. Now consider the bigger triangle, all the angles are of equal angles of 60 each, it is an equilateral triangle, therefore all sides are of equal length. So y is nothing but half of x, that is x=2y