Let C be a circle with centre O. Let AB be a tangent to C such that AB touches the circle at point P. Let X and Y be any two points on C. If $\angle APX = 50^\circ$ and $\angle BPY = 60^\circ$, then find $\angle PXY$ and $\angle PYX$, respectively.

Question

Prepp Team · Accepted Answer

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 $50^\circ$.

Prepp Team · Answer

$50^\circ , 60^\circ$

Prepp Team · Answer

$50^\circ , 130^\circ$

Prepp Team · Answer

$60^\circ, 120^\circ$

Circle Tangent Angle Calculation

Tangent-Chord Theorem Explained

Angle Calculations

Resulting Angles

Important Questions from Circles

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