Circles Class 10: notes and important questions

Tangent and secant

• A line and a circle can have 0 common points (non-intersecting line), 1 common point (tangent) or 2 common points (secant).
• The common point of a tangent and the circle is the point of contact. There is exactly one tangent at each point of a circle.
• A tangent is a limiting case of a secant, when its two points of intersection coincide.
• Number of tangents from a point: inside the circle, 0; on the circle, 1; outside the circle, 2.

Two key theorems (proofs can be asked)

Theorem 1. The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Proof idea: Every point \(Q\) of the tangent other than the contact point \(P\) lies outside the circle, so \(OQ>OP\). Hence \(OP\) is the shortest distance from \(O\) to the line, and the shortest segment from a point to a line is the perpendicular.

Theorem 2. The lengths of the tangents drawn from an external point to a circle are equal.

Proof idea: In \(\triangle OAP\) and \(\triangle OBP\): \(OA=OB\) (radii), \(OP\) is common, and \(\angle OAP=\angle OBP=90^\circ\). By RHS congruence \(PA=PB\). This also gives \(\angle APO=\angle BPO\): the centre lies on the bisector of the angle between the tangents.

Standard consequences

• Tangent length: \(PA=\sqrt{OP^2-r^2}\).
• Angle between tangents and angle at the centre are supplementary: \(\angle APB+\angle AOB=180^\circ\), because \(OAPB\) is a quadrilateral with two right angles.
• Tangents at the ends of a diameter are parallel.
• A quadrilateral \(ABCD\) circumscribing a circle satisfies \(AB+CD=BC+DA\). So a parallelogram circumscribing a circle is a rhombus.
• Incircle of a triangle with sides \(a,b,c\) and semi-perimeter \(s\): the tangent lengths from \(A,B,C\) are \(s-a,\ s-b,\ s-c\). For a right triangle with legs \(a,b\) and hypotenuse \(c\), the inradius is \(r=\dfrac{a+b-c}{2}\).
• In two concentric circles, a chord of the larger circle that touches the smaller one is bisected at the point of contact.

Worked example 1

The tangent from \(P\) to a circle of radius 9 cm has length 12 cm. Find \(OP\).   \(OP=\sqrt{81+144}=15\) cm.

Worked example 2

Tangents \(PA, PB\) make \(\angle APB=40^\circ\). Find \(\angle OAB\).

\(\angle AOB=140^\circ\). Triangle \(OAB\) is isosceles, so \(\angle OAB=\frac{180^\circ-140^\circ}{2}=20^\circ\) (\(=\tfrac12\angle APB\)).

Worked example 3

A circle touches the sides of quadrilateral \(PQRS\) with \(PQ=9\), \(QR=11\), \(RS=8\). Find \(SP\).

\(PQ+RS=QR+SP\Rightarrow SP=9+8-11=6\).

Common errors

• Using \(OP^2=PA^2-r^2\): \(OP\) is the hypotenuse, not a leg.
• Claiming \(\angle APB=\angle AOB\). They are supplementary, not equal.
• Quoting "tangents from an external point are equal" without naming the point. Write "\(PA=PB\) (tangents from \(P\))".

Board-exam tips

• Both theorem proofs are regularly asked for 3 marks. Learn them with a figure, the given, to prove, construction and proof.
• In numerical problems, mark every pair of equal tangents on the figure first. The equations then write themselves.