O is the centre of a circle. From an external point, P two tangents PM and PN have been drawn which touch the circle at M and N. If ∠PON = 50°, then find the value of ∠MPN.

From P we have two tangents PM and PN

We know that if we join point P and centre of circle O then the line PO divides the angle between tangents

⇒ ∠MPO = ∠NPO …(a)

Consider ΔPNO

⇒ ∠PON = 50° …given

As radius ON is perpendicular to tangent PN

⇒ ∠PNO = 90°

Now

⇒ ∠PON + ∠PNO + ∠NPO = 180° …sum of angles of triangle

⇒ 50° + 90° + ∠NPO = 180°

⇒ 140° + ∠NPO = 180°

⇒ ∠NPO = 40° …(i)

From figure

⇒ ∠MPN = ∠MPO + ∠NPO

Using (a)

⇒ ∠MPN = ∠NPO + ∠NPO

⇒ ∠MPN = 2∠NPO

Using (i)

⇒ ∠MPN = 2 × 40°

⇒ ∠MPN = 80°

Hence ∠MPN is 80°