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


Construction: Label the diagram.



In ΔAOP and ΔBOP


OA = OB [Radii of same circle]


OP = OP [Common]


∠OAB = ∠OBP = 90° [Tangent at any point is perpendicular to the radius through point of contact]


ΔAOP ≅ ΔBOP [By Right Angle - Hypotenuse - Side Criteria]


⇒ ∠AOP = ∠BOP [Corresponding parts of congruent triangle are equal]


OP bisects ∠AOB.


i.e. the line joining the centre and the points where the tangents meet bisects the angle between the radii.


More from this chapter

All 26 →
2

In the picture, the small (blue) triangle is equilateral. The sides of the large (red) triangle are tangents to the circumcircle of the small triangle at its vertices.


i) Prove that the large triangle is also equilateral and its sides are double that of the small triangle.


ii) Draw this picture, with sides of the smaller triangle 3 centimetres.


iii) Instead of an equilateral triangle, if we draw the tangents to the circumcircle of any other triangle at its vertices, do we get a similar triangle with double the sides? Justify.

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


 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?