Question
Download Solution PDFABCD is a cyclic quadrilateral such that ∠B = 112°. The tangents at A and C meet at a point P. What is the measure of ∠APC?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
The ∠B = 112°
Formula used:
Opposite angles of the cyclic quadrilateral = 180°
Calculation:
In given cyclic quadrilateral ABCD
⇒ ∠ABC + ∠ADC = 180°
⇒ ∠ADC = 180° - 112° = 68°
Since PA is tangent on the circle at point A and AC is the cord which is subtending the angle ∠D = 68°
The angle between a tangent and a chord of a circle is equal to the angle subtended by the chord in the alternate segment of the circle.
In this case, the tangent at point A (line segment PA) and chord AC subtend ∠PAC in the alternate segment, which is equal to the angle ∠D subtended by the same chord AC. Therefore, ∠PAC = ∠D = 68°.
⇒ ∠PAC = ∠D = 68°
Also,
⇒ ∠PAC = ∠PCA, (As PA and PC are the tangents of A and C)
⇒ ∠PAC = ∠PCA = ∠ADC = 68°
In ΔPAC
⇒ ∠PAC + ∠PCA + ∠APC = 180°
⇒ 68° + 68° + ∠APC = 180°
⇒ ∠APC = 180° - 136° = 44°
∴ The required result will be 44°.
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