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# In the Following Figure, Ab is the Diameter of a Circle with Centre O and Cd is the Chord with Length Equal to Radius Oa.Is Ac Produced and Bd Produced Meet at Point P; Show that ∠Apb = 60° - Mathematics

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#### Question

In the following figure, AB is the diameter of a circle with centre O and CD is the chord with length equal to radius OA.

Is AC produced and BD produced meet at point P; show that ∠APB = 60°

#### Solution

Given – In the figure, AB is the diameter of a circle with centre O. CD is the chord with length equal radius OA. AC and BD produced meet at point P To Prove : ∠APB = 60°
Construction – join OC and OD
Proof – We have CD = OC = OD [given] Therefore, ∆OCD is an equilateral triangle

∴ ∠OCD = ∠ODC = ∠COD = 60°
In ∆AOC, OA = OC [ radii of the same circle]
∴ ∠A = ∠1
Similarly, in ∆BOD, OB = OD [ radii of the same circle]
∴ ∠B = ∠2
Now, in cyclic quad ACBD Since,

∠ACD + ∠B = 180°
[Since sum of opposite angles of a cyclic quadrilateral are supplementary]
⇒ 60° + ∠1+ ∠B = 180°
⇒ ∠1 + ∠B = 180° - 60°
⇒ ∠1 + ∠B = 120°
But, ∠1 = ∠A
∴ ∠A + ∠B = 120°            …..(1)

Now, in ∆APB,
∠P + ∠A + ∠B = 180° [sum of angles of a triangles]
⇒ ∠P +120° = 180°
⇒ ∠P = 180° -120°        [from (1)]
⇒ ∠P = 60° or ∠APB = 60°

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In the Following Figure, Ab is the Diameter of a Circle with Centre O and Cd is the Chord with Length Equal to Radius Oa.Is Ac Produced and Bd Produced Meet at Point P; Show that ∠Apb = 60° Concept: Concept of Circles.
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