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
The sum of the radius and diameter of a circle is 84 cm. What is the circumference of this circle?
The maximum area of a right-angled triangle inscribed in a circle of radius r is
The tangent at a point C of a circle and diameter AB when extended intersect at D, if ∠DCA = 110°, then ∠CBA is equal to
The equation of a circle with diameters are 2x - 3y + 12 = 0 and x + 4y - 5 = 0 and area of 154 sq. units is
The internal center of similitude of two circles $ (x - 1)^2 + (y - 3)^2 = 4 $ and $ (x + 5)^2 + (y - 9)^2 = 16 $ is
