Q1. A tangent PA and PB are drawn from an external point P to a circle. If PA = 8 cm, find PB.
The two tangents from an external point are equal in length.
PA = PB
So PB = 8 cm.
Circles — Class 10
A tangent touches a circle at exactly one point — and that single fact unlocks every tangent problem in this chapter.
TangentA line that touches a circle at exactly one point (the point of contact); a secant cuts it at two points.
Tangent ⊥ radiusThe tangent at any point of a circle is perpendicular to the radius through that point.
Tangents from an external pointThe two tangents drawn from an external point to a circle are equal in length.
Solved examples
Q2. A tangent is drawn to a circle of radius 5 cm from a point 13 cm from the centre. Find the length of the tangent.
The radius to the point of contact is perpendicular to the tangent, forming a right triangle.
Tangent² = (distance)² − (radius)² = 13² − 5² = 169 − 25 = 144
Tangent = √144 = 12 cm
Formula sheet
| Quantity | Formula |
|---|---|
| Number of tangents from a point | on the circle: 1 · outside: 2 · inside: 0 |
| Tangent length | PA = PB (from external point P) |
| Right angle | ∠(tangent, radius) = 90° |
Common mistakes & exam wins
- A radius drawn to the point of contact is always perpendicular to the tangent — this creates a right triangle you can use with Pythagoras.
- The two tangents from the same external point are always equal in length.
- A point inside a circle cannot have any tangent through it.
Frequently asked questions
What is a tangent to a circle?
A straight line that touches the circle at exactly one point, called the point of contact.
What is the relationship between a tangent and the radius?
The tangent at any point is always perpendicular to the radius drawn to that point.
Are the two tangents from an external point equal?
Yes, the lengths of the two tangents drawn from the same external point to a circle are always equal.