CBSE Class 10 Mathematics Circles Notes Set 05

Tangent and Secant

A secant meets a circle in two points; a tangent meets it in exactly one point, called the point of contact. From a point inside the circle no tangent can be drawn, from a point on the circle exactly one, and from an outside point exactly two. A circle has at most two parallel tangents, one at each end of a diameter.
Tangent-radius theorem: the tangent at a point is perpendicular to the radius through that point. Conversely, a line perpendicular to a radius at its outer end is a tangent.

Length of a Tangent

The two tangent lengths from an external point are equal (PT = PT'), and the line from the point to the centre bisects the angle between them. In the right triangle OTP, \( PT^2 = OP^2 - OT^2 \).
If the angle between the two tangents from P is \( 2\theta \), then PT = r cot θ and OP = r cosec θ. Quadrilateral OTPT' is cyclic because the two angles at T and T' are right angles, and the angle between the tangents is supplementary to the angle at the centre.
Tangents from the vertices of a triangle to its incircle give the equal pairs AR = AQ, BP = BR, CP = CQ, so each of AR + BP + CQ and AQ + CP + BR is half the perimeter. For sides a, b, c and semi-perimeter s, the tangent length from A is s − a, from B is s − b, from C is s − c. For the excircle opposite A, the tangent from A is s.

Secants, Chords and Tangent-Chord Angle

If a secant PAB and a tangent PT are drawn from an outside point, then \( PA \times PB = PT^2 \). Two secants from P give \( PA \times PB = PC \times PD \).
The angle between a tangent and a chord through the point of contact equals the angle in the alternate segment. A chord equal to the radius subtends 60° at the centre and 30° at a point on the major arc (150° on the minor arc). A regular hexagon inscribed in a circle of radius r has side r.
Two circles that touch externally have three common tangents; if they intersect in two points they have two; if they touch internally they have one.

Area Related to Circles

Circle: area \( \pi r^2 \) and circumference \( 2\pi r \). A path of width h around a circle of radius r has area \( \pi[(r+h)^2 - r^2] = \pi h(2r+h) \).
Sector of angle θ degrees: area \( \frac{\theta}{360}\pi r^2 \) and arc length \( \frac{\theta}{360}2\pi r \). Segment = sector minus the triangle formed by the radii and the chord. A wheel of diameter d turning n times covers n × πd.
For problems with shaded regions, express the region as sums and differences of simple areas (squares, triangles, sectors, semicircles) and subtract the overlaps once.