Question

PQ is a tangent at P to the circle with centre O and triangle OPQ is isoceles.Find angle OQP​

Answer 1

Answer:

• ∠OQP is 45°

Step-by-step explanation:

Given,

PQ is a tangent at P to the circle with center O.

Triangle OPQ is an isosceles triangle.

We know,

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

So,

• ∠OPQ = 90°

Thus, ∆POQ is right angle triangle. OQ is opposite to 90° so, OQ will be longest side of triangle and it can't be equal to other side.

And,

OPQ is an isosceles triangle, So Side OP and PQ are equal.

Now,

Angles opposite to OP and PQ will also equal.

• ∠POQ = ∠OQP

Sum of all angles of triangle is 180°

⇒ ∠OPQ + ∠OQP + ∠POQ = 180°

⇒ 90° + ∠OQP + ∠OQP = 180°

⇒ 2∠OQP = 180° - 90°

⇒ 2∠OQP = 90°

⇒ ∠OQP = 90°/2

⇒ ∠OQP = 45°

∴ ∠OQP is 45°.

Answer 2

_____________________

• pq is tangent and Oq is radius

we know that tangent is perpendiculatr to radius

• Angle Opq = 90°

Angle OPQ is isoceles

• OP = PQ

Angle OQP = Angle POQ

SO,

angle Oqp = 1/2 X 90°

= 45°

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hope it helps