Class 10 Mathematics · Chapter 10 NotesCircles

When a circle and a line are given in a plane, exactly three situations are possible. The line may have no common point with the circle, in which case it is called a non-intersecting line. The line may meet the circle at two distinct points A and B, and then it is called a secant of the circle. Or the line may meet the circle at only one point A, and then it is called a tangent to the circle. There is no fourth possibility, because a line can cut a circle in at most two points. The common point of a tangent and the circle is called the point of contact, and the tangent is said to touch the circle at that point. A familiar example is a wheel rolling on the ground: the ground line is a tangent to the circle representing the wheel.

• Non-intersecting lineno common point with the circle
• Secanta line intersecting the circle at two points
• Tangenta line intersecting the circle at exactly one point
• Point of contactthe single common point of a tangent and the circle

Imagine a straight wire fixed at a point P of a circular wire so that it can rotate about P. In most positions the straight wire cuts the circle at P and at another point Q. As the wire rotates, Q moves closer and closer to P, and in one particular position the two points of intersection merge into the single point P — that position is the tangent at P. Rotating further produces a new second point of intersection on the other side. Similarly, if you draw lines parallel to a given secant on either side of it, the chords they cut become shorter and shorter until, at two positions, the chord length becomes zero. Those two positions are tangents parallel to the given secant. So a tangent is the secant when the two end points of its corresponding chord coincide, and there cannot be more than two tangents parallel to a given secant.

• A tangent is the limiting position of a secant when the two ends of the chord coincide
• At most two tangents can be drawn parallel to a given secant
• The word tangent comes from the Latin word tangere, meaning to touch

Theorem 10.1 states that the tangent at any point of a circle is perpendicular to the radius through the point of contact. To prove it, take a circle with centre O and a tangent XY touching the circle at P. Choose any other point Q on XY and join OQ. Since XY touches the circle at only one point, Q must lie outside the circle, so OQ is greater than the radius OP. This is true for every point of XY other than P, which means OP is the shortest distance from O to the line XY. The shortest distance from a point to a line is measured along the perpendicular, so OP is perpendicular to XY. Two useful consequences follow: at any point of a circle there is one and only one tangent, and the line containing the radius through the point of contact is sometimes called the normal to the circle at that point.

• Theorem 10.1: the tangent at any point of a circle is perpendicular to the radius through the point of contact
• At any point on a circle, there is exactly one tangent
• The line containing the radius through the point of contact is called the normal to the circle at that point

The number of tangents that can be drawn to a circle from a point depends on where the point lies. If the point is inside the circle, every line through it cuts the circle at two points, so no tangent can be drawn. If the point lies on the circle, exactly one tangent can be drawn through it, as Theorem 10.1 shows. If the point lies outside the circle, exactly two tangents can be drawn from it. In the outside case, if P is the external point and T₁ and T₂ are the points of contact, then the segments PT₁ and PT₂ are called the lengths of the tangents from P to the circle. These two lengths are always equal, which is the content of the next theorem.

• Point inside the circleno tangent
• Point on the circleexactly one tangent
• Point outside the circleexactly two tangents
• Length of the tangent from P is the length of the segment from P to the point of contact

Theorem 10.2 states that the lengths of tangents drawn from an external point to a circle are equal. The proof uses two right triangles. Let O be the centre, P an external point, and PQ, PR the two tangents with points of contact Q and R. Join OP, OQ and OR. By Theorem 10.1, angles OQP and ORP are right angles. In right triangles OQP and ORP, OQ = OR because both are radii of the same circle, and OP is common. So the two triangles are congruent by the RHS rule, and therefore PQ = PR. The same result also follows from the Pythagoras theorem, since PQ² = OP² − OQ² = OP² − OR² = PR². A further observation is that angle OPQ equals angle OPR, so OP bisects the angle between the two tangents; that is, the centre lies on the bisector of the angle between the two tangents.

• Theorem 10.2: the lengths of tangents drawn from an external point to a circle are equal
• Proof uses RHS congruence of triangles OQP and ORP
• Alternative proofPQ² = OP² − OQ² = OP² − OR² = PR²
• The line joining the centre to the external point bisects the angle between the two tangents

The two theorems are used together with earlier results about chords. In two concentric circles, the chord of the larger circle that touches the smaller circle is bisected at the point of contact: the radius to the point of contact is perpendicular to the chord, and the perpendicular from the centre of a circle to a chord bisects the chord. In another standard situation, two tangents TP and TQ are drawn from an external point T to a circle with centre O. Since TP = TQ, triangle TPQ is isosceles, and using the right angle between the radius and the tangent at P, one can show that angle PTQ = 2 × angle OPQ. Problems often combine these facts with the Pythagoras theorem or with similarity of right triangles to find unknown lengths, as when a chord of length 8 cm in a circle of radius 5 cm has tangents at its ends meeting at a point T.

• Concentric circlesthe chord of the larger circle touching the smaller circle is bisected at the point of contact
• If TP and TQ are tangents from T, then TP = TQ and angle PTQ = 2 × angle OPQ
• The perpendicular from the centre to a chord bisects the chord

Tangent properties extend naturally to figures drawn around a circle. If a quadrilateral ABCD circumscribes a circle, so that all four sides are tangents, then the sums of the lengths of opposite sides are equal: AB + CD = AD + BC. This follows by writing each side as the sum of two tangent lengths from its endpoints and using the equality of tangents from an external point. A parallelogram that circumscribes a circle must be a rhombus, because the equal opposite sides together with the tangent-sum relation force all four sides to be equal. Similarly, a triangle can be drawn to circumscribe a circle, and the tangent lengths from each vertex, together with the radius drawn to each point of contact, allow unknown sides to be calculated.

• Quadrilateral ABCD circumscribing a circleAB + CD = AD + BC
• A parallelogram circumscribing a circle is a rhombus
• In a circumscribed triangle, tangent lengths from each vertex are equal