CBSE Class 10 Maths Chapter 10 Circles Notes

Key Theorems and Properties of Circles

• A circle is the locus of all points in a plane that are equidistant from a fixed point (the center).
• A tangent to a circle is a line that intersects the circle at exactly one point, called the point of contact.
• Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact. This implies that the angle between the radius and the tangent at the point of contact is 90°.
• Theorem 10.2: The lengths of tangents drawn from an external point to a circle are equal. If PA and PB are tangents from an external point P to a circle with centre O, then PA = PB.
• Only one tangent can be drawn to a circle at any point on the circle.
• No tangent can be drawn to a circle from a point lying inside the circle.
• Exactly two tangents can be drawn to a circle from a point lying outside the circle.
• The common point of the tangent and the circle is called the point of contact.
• A secant is a line that intersects the circle at two distinct points.

Essential Terms for Circles

Circle

A set of all points in a plane that are at a fixed distance (radius) from a fixed point (center).

Radius

A line segment connecting the center of a circle to any point on its circumference.

Chord

A line segment connecting any two points on the circumference of a circle.

Diameter

A chord that passes through the center of the circle. It is the longest chord and is twice the radius (D = 2r).

Tangent

A line that touches the circle at exactly one point. This point is known as the point of contact.

Secant

A line that intersects a circle at two distinct points.

Point of Contact

The single common point shared by a tangent line and the circle.

Concentric Circles

Two or more circles that share the same center but have different radii.

Understanding Tangents and their Properties

The concept of tangents to a circle is central to Class 10 geometry. A tangent is fundamentally different from a secant. While a secant cuts through a circle at two points, a tangent just 'kisses' the circle at a single unique point. This point is called the point of contact. Imagine a wheel rolling on a flat surface; the point where the wheel touches the ground at any instant is the point of contact, and the ground itself acts as a tangent line.

The most critical property of a tangent is described by Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact. This means if you draw a radius from the center of the circle to the point where the tangent touches it, the angle formed between this radius and the tangent line will always be 90 degrees. This property is extremely useful in solving problems involving right-angled triangles, especially when finding lengths or angles. For instance, if you have a right angle, you can often apply the Pythagoras theorem (hypotenuse² = base² + perpendicular²) or trigonometric ratios.

Another vital theorem is Theorem 10.2: The lengths of tangents drawn from an external point to a circle are equal. If you pick any point outside a circle, you can draw exactly two tangents from it to the circle. This theorem states that the segments of these two tangents from the external point to their respective points of contact on the circle will have identical lengths. For example, if point P is external and PA and PB are tangents to the circle, then PA = PB. This property is frequently used in problems that ask for the perimeter of a polygon circumscribing a circle, or when proving properties about angles formed by these tangents and the center. The line segment joining the external point to the center of the circle bisects the angle between the two tangents and also bisects the angle between the radii drawn to the points of contact. Understanding these relationships is key to excelling in this chapter.

Tangent vs. Secant: Key Differences

Solved Examples

• Example 1: A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Find the length of PQ. Solution: Since the tangent at any point of a circle is perpendicular to the radius through the point of contact, OP ⊥ PQ. Thus, ΔOPQ is a right-angled triangle at P. By Pythagoras Theorem: OQ² = OP² + PQ². Given OQ = 12 cm, OP (radius) = 5 cm. So, 12² = 5² + PQ² => 144 = 25 + PQ² => PQ² = 119 => PQ = √119 cm.
• Example 2: From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. Find the radius of the circle. Solution: Let the point of contact be P, and the center be O. Then QP = 24 cm and OQ = 25 cm. We know that the radius OP is perpendicular to the tangent QP. So, ΔOPQ is a right-angled triangle at P. By Pythagoras Theorem: OQ² = OP² + PQ². 25² = OP² + 24² => 625 = OP² + 576 => OP² = 625 - 576 = 49 => OP = √49 = 7 cm. The radius of the circle is 7 cm.

Exam Tip: Avoiding Common Mistakes

Many students forget that Theorem 10.1 (radius ⊥ tangent) means forming a right-angled triangle is almost always the first step in problems involving tangents and radii. Always draw a clear diagram and mark the 90° angle. For problems using Theorem 10.2 (equal tangents from external point), remember that the line joining the external point to the center bisects the angle between the tangents and also the angle subtended by the points of contact at the center. Look out for questions combining circle theorems with properties of quadrilaterals or triangles (e.g., area calculations, congruency/similarity proofs).

Quick Revision Check

• Q: How many tangents can be drawn to a circle from a point lying on the circle? A: Exactly one tangent can be drawn to a circle from a point lying on the circle.
• Q: If a line intersects a circle at two distinct points, what is it called? A: It is called a secant.
• Q: What is the angle between the radius and the tangent at the point of contact? A: The angle is always 90 degrees (perpendicular).
• Q: Two tangents from an external point P to a circle are PA and PB. If PA = 10 cm, what is the length of PB? A: PB = 10 cm, according to Theorem 10.2 (lengths of tangents from an external point are equal).

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