Question

oc radius equal to chord cd and ab is diameter and ac and bd produced meet at p so prove that angle cpd = 60°​

Answer 1

Answer:

cpd = 60⁰

Step-by-step explanation:

AB is a diameter of the circle with centre O and chord CD is equal to radius OC. AC and BD produced meet at P.

To Prove : ∠CPD = 600

Construction : Join AD.

Proof :

In ΔOCD,

OC = OD ...(i) [Radii of the same circle]

OC = CD ....(ii) [Given]

From (i) and (ii),

OC = OD = CD ∴ DOCD is equilateral

∴ ∠COD = 600

∴ ∠CAD = 1/2∠COD = 1/2∠(600) = 300 [∵ Angle subtended by any arc of a circle at the centre is twice the angle subtended by it at any point of the reaming part of the circle]

=>∠PAD = 300 .....(iii) And,

∠ADB = 900 .....(iv) [Angle in a semi-circle]

=> ∠ADB + ∠ADP = 1800 [Linear Pair Axiom]

=> 900 + ∠ADP = 1800

[From (iv)]

=> ∠ADP = 900...(v)

In ΔDP, ∠ADP + ∠PAD +∠ADP = 1800 [∵The sum of the three angles of a triangles is 180º]

=> ∠APD + 300+ 900 = 1800 [From (iii) and (v)]

=> ∠APD + 1200 = 1800 =>∠APD = 1800 - 1200 = 600

=> ∠CPD = 600.

Explanation:

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Answer 2

the correct answer is cpd = 60°