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Can you help me about the title? I don't know what's appropriate for a geometry problem!

In the following figure find measure of angle $x$.

enter image description here

I wrote sin law two times. One in ABC like

$$\frac{8+BD}{\sin 120}=\frac{4}{\sin x}$$

and one in ABD like this

$$\frac{BD}{\sin 30}=\frac{4}{\sin (\angle{BDA})}$$

Now if you eliminate $BD$ and use $\angle{BDA}=90+x$, a trigonometric equation comes out. Solving gives $x=20$ degrees.

I feel that there is a neat way around of doing this. Can you help?

Ghartal
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    While it's not necessarily tied to this, I feel like - from an experience standpoint, not any sort of hardcore proof or anything - that some of the "neat" ways tied to this would relate to the angle being some sort of "special" one. Like how $45^\circ$ has those nice values in the various trig functions. There are a lot (https://en.wikipedia.org/wiki/Trigonometric_constants_expressed_in_real_radicals), but $20^\circ$ apparently isn't, and it may be tied to the 20-70-90 triangle not being constructible. While I'm not going to claim it as foolproof, I think this would hint at it being unlikely. – PrincessEev Nov 16 '18 at 10:29
  • Thanks. I agree with you. – Ghartal Nov 16 '18 at 10:40
  • @Ghartal,I computed x=20° with help of equations solver.How to compute it manually? – Win_odd Dhamnekar Nov 16 '18 at 11:03
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    @DhamnekarWinod you get $$8+\frac{2}{\cos x}=\frac{2\sqrt{3}}{\sin x}$$ arranging gives $$\frac{\sqrt{3}}{2} \cos x -\frac{1}{2} \sin x=2 \sin x \cos x$$ Thus $$\sin(60-x)=\sin 2x$$ and $x=20$ – Ghartal Nov 16 '18 at 11:14

1 Answers1

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Let $I$ be the midpoint of $[DC]$ then $AI=\frac{1}{2} DC=4$ (since $ADC$ is a right triangle)

Now you get that, $ABI$ is an isoscles triangle of vertex $A$ which gives you that $\angle ABI =\angle AIB=2x$

(Because if you draw a circle $(C)$ circumscribed about triangle $ACD$ you'll notice that $\angle AID$ is a central angle which is double $\angle ACD$ the interior angle in $(C)$ facing the same arc $AD$)

Finally by summing up the angles in triangle $ABC$ you'll get that $2x+x+120=180$ which gives you that $x=20^\circ$

enter image description here

Fareed Abi Farraj
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