Question

The angle subtended by an arc at the centre is 104° then find the value of angle subtended by the same arc at any point on the remaining part of the circle. *​

Answer 1

Answer:

"Proof:

Consider a circle with centre “O”. Here the arc PQ of the circle subtends angle POQ at Centre O and ∠PAQ at a point A on the remaining part of the circle.

To prove: ∠POQ = 2∠PAQ.

To prove this, join AO and extend it to point B.

There are two general cases while proving this theorem.

Step-by-step explanation:

Consider a triangle APO,

Here, OA = OP (Radii)

Since, the angles opposite to the equal sides are equal,

∠OPA = ∠OAP …(1)

Also, by using the exterior angle property (exterior angle is the sum of interior opposite angles),

We can write,

∠BOP = ∠OAP + ∠OPA

By using (1),

∠BOP = ∠OAP + ∠OAP

∠BOP = 2∠OAP… (2)

Similarly, consider another triangle AQO,

OA = OQ (Radii)

As the angles opposite to the equal sides are equal,

∠OQA = ∠OAQ … (3)

Similarly, by using the exterior angle property, we get

∠BOQ = ∠OAQ + ∠OQA

∠BOQ = ∠OAQ + ∠OAQ (using (3))

∠BOQ = 2∠OAQ …(4)

Adding (2) and (4) we get,

∠BOP + ∠BOQ = 2∠OAP + 2∠OAQ

∠POQ = 2(∠OAP + ∠OAQ)

∠POQ = 2∠PAQ.

Hence, case (1) is proved. "