Circles — Study Notes

1. Introduction

A circle is the set of all points in a plane that are at a fixed distance from a fixed point called the centre. This chapter focuses on the geometry of circles, particularly the properties of tangents to a circle. A tangent is a line that touches the circle at exactly one point, called the point of contact, and it plays a central role in the study of circles.

The chapter begins by classifying the possible positions of a line relative to a circle: the line may be a secant, cutting the circle at two points, or a tangent, touching it at exactly one point. The main results concern the tangent at a point: it is perpendicular to the radius at the point of contact, and the lengths of tangents drawn from an external point are equal.

These theorems are proved rigorously in the chapter and are then used to solve numerical problems involving lengths of tangents and radii. A clear understanding of the tangent properties and their proofs is essential, as theorem-based questions and numerical applications both appear frequently in board examinations.

2. Basic Terms and Positions of a Line

Terms

• Radius: the fixed distance from the centre to any point on the circle.
• Secant: a line that intersects the circle at two distinct points.
• Tangent: a line that intersects the circle at exactly one point, the point of contact.
• Point of contact: the unique point where a tangent touches the circle.

Positions of a Line Relative to a Circle

A line and a circle can have three possible relationships: 1. No intersection: the line does not meet the circle (distance from centre to line > radius). 2. Tangent: the line touches the circle at exactly one point (distance from centre to line = radius). 3. Secant: the line cuts the circle at two points (distance from centre to line < radius).

A tangent to a circle is a special case of a secant where the two intersection points coincide.

3. Tangent Perpendicular to Radius

Theorem

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

In symbols, if OP is the radius to the point of contact P and the tangent at P is line l, then OP is perpendicular to l.

Proof Idea

Suppose the tangent is not perpendicular to the radius. Then a perpendicular can be drawn from the centre O to the line, meeting it at a point Q different from P. Since OQ is the shortest distance from O to the line, OQ < OP. But OP equals the radius, and any other point of the line would then be closer than the radius distance, implying the line meets the circle in another point, contradicting the fact that it is a tangent. Hence the tangent must be perpendicular to the radius.

Consequences

• From a point on the circle, exactly one tangent can be drawn.
• The radius at the point of contact is the perpendicular from the centre to the tangent.

4. Tangents from an External Point

Theorem

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

If PA and PB are two tangents drawn from an external point P to a circle with centre O, touching at A and B, then: PA = PB

Proof Idea

Consider the triangles OAP and OBP. OA = OB (radii), OP is common, and angles OAP and OBP are right angles because each tangent is perpendicular to the radius at the point of contact. By the RHS congruence rule, triangle OAP is congruent to triangle OBP, so PA = PB.

Additional Properties

• The line joining the centre to the external point bisects the angle between the two tangents, i.e., angle AOP = angle BOP.
• In quadrilateral PAOB, the angles at A and B are right angles, so angle APB + angle AOB = 180 degrees.

These results are used to find lengths, angles, and to construct proofs in circle problems.

5. Numerical Problems

Typical problems combine the tangent theorems with Pythagoras' theorem.

Example: A tangent PA of length 8 cm is drawn from an external point P to a circle of radius 6 cm. Find the distance OP. Since OA is perpendicular to PA, triangle OAP is right angled at A. OP^2 = OA^2 + PA^2 = 36 + 64 = 100 OP = 10 cm.

This pattern, right triangle + Pythagoras, is the backbone of most numerical tangent problems.

Tangent Length in Common Configurations

In a quadrilateral circumscribing a circle (all four sides tangent), the sums of opposite sides are equal: AB + CD = AD + BC

This property is derived by labelling the tangent segments from each vertex and using the equality of tangent lengths from a common external point.

6. Proof-Based Problems

Many exam questions ask students to prove a property using the two theorems above. The standard approach:

1. State the given and what is to be proved clearly.
2. Draw radii to the points of contact and join the centre to the external point.
3. Apply the tangent-perpendicular-radius theorem to get right angles.
4. Use congruence (usually RHS) or equal tangent lengths to complete the proof.

Problems include proving that tangents at the ends of a diameter are parallel, that the tangents at the ends of a chord are equal in specific configurations, and that certain quadrilaterals are cyclic based on tangent properties.

Quick Revision Tables

Table 1: Line-Circle Relationships

Table 2: Key Theorems and Properties

Mind Map

Important Diagrams (SVG)

Diagram 1: Tangent Perpendicular to Radius

Diagram 2: Equal Tangents from an External Point

7. Solved Example Approach

The standard way to solve numerical problems about tangents is to identify the right triangle, apply the tangent-perpendicular-radius theorem to locate the right angle, and then use Pythagoras' theorem. Consider a circle with centre O and radius 5 cm, from which a tangent PA of length 12 cm is drawn from an external point P. Since the radius OA is perpendicular to the tangent PA at the point of contact A, the triangle OAP is right angled at A. The hypotenuse is OP, because it joins the centre to the external point, while OA and PA are the two legs. Applying Pythagoras' theorem, OP squared equals OA squared plus PA squared, which is 25 + 144 = 169, so OP equals 13 cm. The most common mistake in such problems is treating the tangent itself as the hypotenuse; drawing the triangle and marking the right angle before writing the equation prevents this error.

A second common configuration asks for the length of a tangent when the distance from the centre and the radius are given. Suppose a point P is 10 cm from the centre O of a circle of radius 6 cm, and PA is a tangent touching the circle at A. Again triangle OAP is right angled at A, so PA squared equals OP squared minus OA squared, which is 100 - 36 = 64, giving PA = 8 cm. Notice that this is the reverse of the previous example: when we know the hypotenuse and one leg, we subtract rather than add. Recognising which side is known and which is required is the key to choosing the correct form of Pythagoras' theorem.

A third type of problem involves tangents from an external point used together with the property of equal tangent lengths. When two tangents PA and PB are drawn from P to the circle, we know immediately that PA = PB, because the triangles OAP and OBP are congruent by the RHS rule. This equality, combined with the fact that the radii OA and OB are equal, allows us to find perimeters and side lengths in more complex figures, such as quadrilaterals circumscribing a circle. For such a quadrilateral, the sum of the lengths of one pair of opposite sides equals the sum of the other pair, a result that follows by labelling each tangent segment and using the equality of tangents drawn from each vertex. In a proof-based question, the same reasoning is applied in reverse: students draw the radii to the points of contact, mark the right angles, and then use congruence to establish the required equality. Mastering these three patterns, the right triangle, the equal tangents, and the circumscribed quadrilateral, covers almost every numerical and proof question asked from this chapter.

Common Mistakes

1. Calling a line that cuts the circle at two points a tangent; such a line is a secant. A tangent touches at exactly one point.
2. Using the tangent-radius theorem without stating which radius, i.e., forgetting that the radius must be drawn to the point of contact.
3. Applying Pythagoras in the wrong triangle, e.g., treating the tangent length as the hypotenuse instead of a leg.
4. Forgetting that a quadrilateral circumscribing a circle satisfies AB + CD = AD + BC, not AB + BC = CD + DA.
5. Concluding that tangents from a point inside the circle are equal; tangents can only be drawn from an external point.
6. In angle problems, forgetting that the line from the centre to the external point bisects the angle between the tangents.
7. Stating that a tangent is perpendicular to the chord instead of the radius at the point of contact.

Exam Tips

1. State the tangent-radius theorem explicitly when using a right angle in a numerical problem.
2. Always draw radii to the points of contact and mark the right angles before writing any equations.
3. In numerical questions, identify the right triangle and write the Pythagoras relation before substituting numbers.
4. For theorem proofs, use RHS congruence between the two triangles formed by the centre and the external point.
5. In quadrilateral circumscribing problems, mark the equal tangent segments from each vertex with letters and add them.
6. Practise the result that tangents at the ends of a diameter are parallel, a common short proof question.
7. Remember that from an external point exactly two tangents can be drawn, and from a point on the circle exactly one.

Conclusion

Circles offer some of the most elegant results in elementary geometry, and the tangent properties form the heart of this chapter. The perpendicularity of the tangent to the radius at the point of contact and the equality of tangents from an external point are simple yet powerful facts that unlock a wide range of numerical and proof-based problems. Combining these theorems with Pythagoras' theorem allows students to solve length and distance problems efficiently. A mastery of tangent properties not only secures high marks in the board examination but also builds the geometric intuition needed for higher studies in conic sections and calculus.