Q3 of 26 Page 166

The picture shows the tangents at two points on a circle and the radii through the points of contact.

Prove that this line is the perpendicular bisector of the chords joining the points of contact.


Let us label the diagram



To Prove : OP is a perpendicular bisector of AB.


In ΔAOF and ΔBOF


OA = OB [Radii of same circle]


∠AOF = ∠BOF [i.e. the line joining the centre and the points where the tangents meet bisects the angle between the radii]


OF = OF [Common]


⇒ ΔAOF ≅ ΔBOF [By Side-Angle-Side Criterion]


⇒ AF = BF and ∠AFO = ∠BFO [Corresponding parts of congruent triangles are equal]


Also,


∠AFO + ∠BFO = 180° [Linear Pair]


⇒ ∠AFO + ∠AFO =180°


⇒ ∠AFO = 90°


⇒ OP ⊥ AB


And OP bisects AB


⇒ OP is perpendicular bisector of AB.


More from this chapter

All 26 →
3

The picture shows the tangents at two points on a circle and the radii through the points of contact.

Prove that the tangents have the same length.


 3

The picture shows the tangents at two points on a circle and the radii through the points of contact.

Prove that the line joining the centre and the points where the tangents meet bisects the angle between the radii.


 4

Prove that the quadrilateral with sides as the tangents at the ends of a pair of perpendicular chords of a circle is cyclic.

What sort of a quadrilateral do we get if one chord is a diameter? And if both chords are diameters?


 1

In the picture, the sides of the large triangle are tangents to the circumcircle of the small triangle, through its vertices.


Calculate the angles of the large triangle.