Circles (NCERT Ch. 9)

• Understand the tangent to a circle and its key property.
• Prove and apply: the tangent at any point of a circle is perpendicular to the radius through the point of contact.
• Use the property that lengths of tangents drawn from an external point to a circle are equal.

1. A tangent touches the circle at exactly one point (the point of contact), while a secant intersects it at two points.
2. Key theorem: The tangent at any point of a circle is perpendicular to the radius at the point of contact.
3. Key theorem: Lengths of tangents drawn from an external point to a circle are equal.
4. These two theorems together are the main tools for solving most Class 10 circle-tangent problems.

From an external point P, two tangents PA and PB are drawn to a circle with centre O. If PA = 8 cm, find PB.

Tangents from the same external point to a circle are equal in length.
So PB = PA = 8 cm.

• Forgetting that the tangent-radius angle is always exactly 90°.
• Confusing a tangent (touches at 1 point) with a chord (joins 2 points on the circle) or secant (extends through 2 points).
• Assuming tangent lengths are equal from ANY point, not just from the same external point.

• If a radius is 5 cm and the tangent-radius angle is given, what is that angle?
• From external point P, PA = 6 cm is one tangent. What is the length of the other tangent PB?

Open the Practice tab for graded questions on Circles (NCERT Ch. 9).