Pre-RMO -bash 1

Geometry Level 4

The circle ω touches the circle Ω internally at P. The centre O of Ω is outside ω. Let XY be a diameter of Ω which is also tangent to ω. Assume P Y > P X. Let P Y intersect ω at Z. If Y Z = 2P Z ,what is the magnitude of ∠P Y X in degrees?


The answer is 15.

Abhishek Alva by abhishek alva , 19, India

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1 solution

Marta Reece
Jun 10, 2017

Triangles △ Y O P \triangle YOP and △ Z Q P \triangle ZQP are similar, therefore from the fact that Y Z ‾ = 2 Z P ‾ \overline{YZ}=2\overline{ZP} we can derive relationship between the radii R R , of the large circle, and r r , of the small one.

Y P ‾ = 3 Z P ‾ ⟹ R = O P ‾ = 3 Q P ‾ = 3 r \overline{YP}=3\overline{ZP}\implies R=\overline{OP}=3\overline{QP}=3r

sin ⁡ ∠ Q O S = r 2 r = 1 2 ⟹ ∠ Q O S = 3 0 ∘ \sin \angle QOS=\dfrac{r}{2r}=\dfrac12\implies\angle QOS=30^\circ

∠ P Y X = 1 2 ∠ P O X = 1 5 ∘ \angle PYX=\dfrac12\angle POX=\boxed{15^\circ}

5 Helpful 0 Interesting 0 Brilliant 0 Confused

Did the same way.

Niranjan Khanderia - 3 years, 11 months ago

A general fact. The line P S PS meets the bigger circle at midpoint of arc X Y XY for any chord Y X YX .

Vishwash Kumar ΓΞΩ - 3 years, 11 months ago

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