Question

In the given figure. O is the centre of the circle. Determine APC, if DA and DC are tangents and
ADC = 50​

Answer 1

In the given figure. O is the centre of the circle.

Given,

DA and DC are tangents

ADC = 50​

Construction: Join OC and OA.

as we know that, the tangents are perpendicular to the radius at the point of contact, we have,

OC = OA = radius of circle

DA and DC are tangents

∴ DA ⊥ OA and DC ⊥ OC

⇒ ∠ DAO = ∠ DCO = 90°

Now consider, quadrilateral DAOC,

as we know that, sum of interior angles of a quadrilateral is equal to 360°, we get,

∠ A + ∠ D + ∠ O + ∠ C = 360°

90° + 50° + ∠ O + 90° = 360°

∠ O + 230° = 360°

∠ O = 360° - 230°

∴ ∠ O = 130°

Central angle theorem states that, central angle from 2 chosen points on the circle is twice the inscribed angle from those 2 points.

Using the central angle theorem, we get,

∠ APC = (360° - 130°)/2 = 230°/2

∴ ∠ APC = 115°