Mathematics
In the figure (i) given below, ∠CBP = 40°, ∠CPB = q° and ∠DAB = p°. Obtain an equation connecting p and q. If AC and BD meet at Q so that ∠AQD = 2q° and the points C, P, B and Q are concyclic, find the values of p and q.
Circles
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Answer
From figure,
∠ADC = ∠CBP = 40°. (∵ angles in alternate segments are equal.)
Since sum of angles in a triangle = 180°.
In △ADP,
∠DAP + ∠APD + ∠ADP = 180°
From figure, ∠ADP = ∠ADC = 40°.
⇒ p° + q° + 40° = 180°
⇒ p° + q° = 180° - 40°
⇒ p° + q° = 140° …..(i)
Join AC and BD as shown in the figure below:
∠CQB = ∠AQD = 2q° (∵ vertically opposite angles are equal.)
Given C, P, B, Q are concyclic.
∴ ∠CPB + ∠CQB = 180°
⇒ q° + 2q° = 180°
⇒ 3q° = 180°
⇒ q° = 60°.
Putting value of q in equation (i) we get,
⇒ p° + 60° = 140°
⇒ p° = 140° - 60° = 80°.
Hence, the value of p = 80 and q = 60 and the relation between p and q is p + q = 140.
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