CBSE Class 10 Mathematics Circles Notes Set 06

Tangents to a Circle

A tangent touches a circle at exactly one point; a secant cuts it at two. The tangent at a point is perpendicular to the radius through that point. From an external point P two tangents can be drawn; they are equal in length, subtend equal angles at the centre, and make equal angles with the line from P to the centre. The line OP is the perpendicular bisector of the chord joining the two points of contact, and the angle between the tangents is \( 180^\circ \) minus the angle at the centre between the radii to the points of contact.
The tangent length from a point at distance d from the centre of a circle of radius r is \( \sqrt{d^2-r^2} \).

Circumscribed Figures

If a quadrilateral has an incircle, the sums of opposite sides are equal: \( AB+CD=BC+AD \). A parallelogram that circumscribes a circle must be a rhombus. If a circle touches side BC of a triangle and the extensions of AB and AC, then the tangent length from A equals half the perimeter. For a right triangle with legs a, b and hypotenuse c, the inradius is \( \frac{a+b-c}{2} \). Opposite sides of a circumscribed quadrilateral subtend supplementary angles at the centre.

Chords and Concentric Circles

A chord of the larger of two concentric circles that touches the smaller one is bisected at the point of contact, so its length is \( 2\sqrt{R^2-r^2} \). The two tangents at the ends of a diameter are parallel and the distance between them is the diameter. For parallel tangents l and m and a third tangent meeting them at S and T, the angle SOT is a right angle.