Concept Topic GuideCBSE & ICSEClass 10MathematicsCirclesVerified Faculty Notes & Solutions
Circles – Class 10 Maths Chapter 10: Tangent Properties & Theorems
Class 10 Circles notes: tangent from external point, number of tangents, length of tangent theorem, solved problems for CBSE & ICSE.
4 students readUpdated 4 September 2026
Chapter 10: Circles
10.1 Tangent to a Circle
A tangent to a circle is a line that touches the circle at exactly one point, called the point of tangency (or point of contact).
Number of Tangents from a Point
- From a point inside the circle: 0 tangents
- From a point on the circle: 1 tangent (exactly one)
- From a point outside the circle: 2 tangents
10.2 Important Theorems
Theorem 1: Tangent is Perpendicular to Radius
Statement: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
If OA is the radius and PA is the tangent at point A, then OA ⊥ PA (∠OAP = 90°).
If OA is the radius and PA is the tangent at point A, then OA ⊥ PA (∠OAP = 90°).
Theorem 2: Equal Tangent Lengths from External Point
Statement: The lengths of tangents drawn from an external point to a circle are equal.
If PA and PB are tangents from external point P to a circle with centre O, then PA = PB.
Also: ∠OPA = ∠OPB (OP bisects the angle between the tangents).
If PA and PB are tangents from external point P to a circle with centre O, then PA = PB.
Also: ∠OPA = ∠OPB (OP bisects the angle between the tangents).
10.3 Solved Examples
Example: Two tangents from an external point form an angle of 60°. Find the length of each tangent if the radius is 5 cm.
Solution
Let PA and PB be tangents from point P, and O be the centre. ∠APB = 60°.
Since OP bisects ∠APB: ∠OPA = 30°. Also ∠OAP = 90° (radius ⊥ tangent).
In right △OAP: tan 30° = OA/PA → 1/√3 = 5/PA → PA = 5√3 ≈ 8.66 cm
Indexed Topics & Examination Keywords
#circles class 10#tangent to circle#tangent properties#number of tangents#length of tangent#NCERT maths#CBSE#ICSE
SixBytes Academic Coaching • Shyampur & Premnagar
Aiming for 95%+ in CBSE / ICSE Board & Competitive Exams?
Join daily batches at SixBytes Institute for personalized mentoring, one-on-one doubt solving, and structured mock test series for Classes 9–12 & NDA.