ABCD is a cyclic quadrilateral such that B = 104°. The tangents at A and C meet at a point P. What is the measure of ∠APC?

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ICAR Technician 28 Feb 2022 Shift 2
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  1. 24°
  2. 38°
  3. 28°
  4. 26°

Answer (Detailed Solution Below)

Option 3 : 28°
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Detailed Solution

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Given:

The ∠B = 104°

Formula used:

Opposite angles of the cyclic quadrilateral = 180° 

Calculation:

In given cyclic quadrilateral ABCD

F2 Madhuri Engineering 26.04.2022 D3

⇒ ∠ABC + ∠ADC = 180° 

⇒ ∠ADC = 180° - 104° = 76° 

Since PA is tangent on the circle at point A and AC is the cord which is subtending the angle ∠D = 76°
 

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 = 76°.

⇒ ∠PAC = ∠D = 76°

Also,

⇒ ∠PAC  =  ∠PCA, (As PA and PC are the tangents of A and C) 

⇒ ∠PAC = ∠PCA = ∠ADC = 76°    

In ΔPAC

⇒ ∠PAC + ∠PCA + ∠APC = 180° 

⇒ 76° + 76° + ∠APC = 180° 

⇒ ∠APC = 180° - 152° = 28° 

∴ The required result will be 28°.

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