nice geometry

Geometry Level 3


The answer is 42.

Aly Ahmed by Aly Ahmed , 34, Egypt

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1 solution

Let ∣ A B ‾ ∣ = ∣ B C ‾ ∣ = a , ∣ B D ‾ ∣ = b |\overline {AB}|=|\overline {BC}|=a, |\overline {BD}|=b . Then

a b = sin ⁡ 30 ° sin ⁡ 18 ° = sin ⁡ ( 96 ° − x ) sin ⁡ ( 72 ° − x ) \dfrac{a}{b}=\dfrac{\sin 30\degree}{\sin 18\degree}=\dfrac{\sin (96\degree-x)}{\sin (72\degree-x)}

⟹ tan ⁡ x = 2 sin ⁡ 18 ° sin ⁡ 96 ° − sin ⁡ 72 ° 2 sin ⁡ 18 ° cos ⁡ 96 ° − cos ⁡ 72 ° = tan ⁡ 42 ° \implies \tan x=\dfrac{2\sin 18\degree\sin 96\degree-\sin 72\degree}{2\sin 18\degree\cos 96\degree-\cos 72\degree}=\tan 42\degree .

Hence x = 42 ° x=\boxed {42\degree} .

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