Question

In the given figure, O is the centre of the circle. Seg AB, seg AC are tangent segments. Radius of the circle is r and l(AB) = r , Prove that, ▢ABOC is a square.

Answer 1

O is the centre of the circle. Seg AB and seg AC are tangent segments drawn from A to the circle.

Join OB and OC.



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

∴ ∠OBA = ∠OCA = 90º

Now, OB = OC = r .....(1) (Radii of the circle)

Tangent segments drawn from an external point to a circle are congruent.

∴ AC = AB = r .....(2)

From (1) and (2), we have

AB = OB = OC = AC

In quadrilateral ABOC,

AB = OB = OC = AC and ∠OBA = ∠OCA = 90º

∴ Quadrilateral ABOC is a square. (All sides of square are equal and measure of each angle is 90º)