In the adjoining 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

Question

In the adjoining figure, O is the center 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

Given: O is the centre of circle. 

seg AB and seg AC are the tangents, radius = r, /(AB) = r. 

To prove: □ABOC is a square. 

Construction: Draw seg OB and seg OC. 

Proof: 

seg AB and seg AC are the tangents to the circle. [Given] 

∴ AB = AC [Tangent segment theorem] 

But, AB = r [Given] 

∴ AB = AC = r ……….. (i) 

Also, OB = OC = r ……….. (ii) [Radii of the same circle]

∴ AB = AC = OB = OC [From (i) and (ii)] 

∴ □ABOC is a rhombus. 

∠OBA = 90° [Tangent theorem] 

∴ □ABOC is a square [A rhombus is a square, if one of its angles is a right angle]