Circle Tangent Angle Calculation
This solution utilizes the Angle Between Tangent and Chord Theorem to determine the required angles.
Tangent-Chord Theorem Explained
The Angle Between Tangent and Chord Theorem establishes that the angle formed by a tangent and a chord drawn from the point of contact is equivalent to the angle subtended by the chord in the alternate segment of the circle.
Angle Calculations
- Given: AB is a tangent to circle C at point P.
- Given: $\angle APX = 50^\circ$. This angle is formed between the tangent AB and the chord PX.
- As per the Tangent-Chord Theorem, the angle subtended by chord PX at the circumference in the alternate segment (at point Y) is equal to $\angle APX$.
- Therefore, $\angle PYX = \angle APX = 50^\circ$.
- Given: $\angle BPY = 60^\circ$. This angle is formed between the tangent AB and the chord PY.
- As per the Tangent-Chord Theorem, the angle subtended by chord PY at the circumference in the alternate segment (at point X) is equal to $\angle BPY$.
- Therefore, $\angle PXY = \angle BPY = 60^\circ$.
Resulting Angles
The question requires finding $\angle PXY$ and $\angle PYX$ respectively.
- $\angle PXY = 60^\circ$
- $\angle PYX = 50^\circ$
The respective angles are $60^\circ$ and $50^\circ$.
Important Questions from Circles
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The inner circumference of a circular race track 14 cm wide is 440 cm. Find the radius of the outer circle.
