Circles | Class 10 Mathematics Notes
By ConceptScroll Team · Published on 17 July 2026 · 3 min read
Circles – this guide gives you a concise, exam-ready overview of Circles from Class 10 Mathematics, written by ConceptScroll editors and reviewed against the latest NCERT textbook.
Tangent to a Circle
A tangent to a circle is a straight line that touches the circle at exactly one point. This point is called the point of contact. Unlike a secant, which intersects the circle at two points, a tangent just 'grazes' the circle, not crossing it. The tangent line lies in the plane of the circle and does not intersect the circle at any other point. The existence of a tangent at a point on the circle can be understood by considering the position of a line relative to the circle: when the line moves closer to the circle, it first intersects at two points (secant), then at exactly one point (tangent), and finally no points (external line).
📊 Diagram: Fig. 10.3 (i); Fig. 10.3 (ii)
🧪 Activity: Using a compass and ruler, draw a circle and try to draw a tangent line at a point on the circle by carefully positioning the ruler.
🔗 Connection: This section introduces tangents, leading to the exploration of their properties in the next section.
Frequently asked questions
1. How many tangents can a circle have?
A circle can have infinitely many tangents. At every point on the circle, there is exactly one tangent line. Since a circle has infinitely many points, it has infinitely many tangents.
2. Fill in the blanks : (i) A tangent to a circle intersects it in point (s). (ii) A line intersecting a circle in two points is called a . (iii) A circle can have parallel tangents at the most. (iv) The common point of a tangent to a circle and the circle is called .
(i) A tangent to a circle intersects it in point (s): one point. (ii) A line intersecting a circle in two points is called a secant. (iii) A circle can have parallel tangents at the most: two. (iv) The common point of a tangent to a circle and the circle is called point of contact.
3. 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. Length PQ is : (A) 12 cm (B) 13 cm (C) 8.5 cm (D) 119 cm.
Given: Radius OP = 5 cm, OQ = 12 cm, PQ is tangent at P. Since PQ is tangent at P, OP ⊥ PQ. In right triangle OPQ, by Pythagoras theorem: PQ² + OP² = OQ² => PQ² = OQ² - OP² = 12² - 5² = 144 - 25 = 119 => PQ = √119 ≈ 10.91 cm None of the options exactly match 10.91 cm, but closest is (C) 8.5 cm is less, (B) 13 cm is close. Rechecking the problem setup: Possibly OQ is the distance from centre to Q on the line through O and tangent at P. Actually, since PQ is tangent at P, and Q lies on the line th
4. Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.
To solve this, first draw a circle. Then draw a given line. Draw one line parallel to the given line such that it touches the circle at exactly one point (tangent). Draw another line parallel to the given line such that it intersects the circle at two points (secant). This satisfies the condition.
Ready to ace this chapter?
Get the full Circles chapter — interactive notes, diagrams, worked solutions, polls and a free practice quiz — in the ConceptScroll app.
Study smarter with ConceptScroll
Daily NCERT-aligned reels, AI doubt solving and chapter quizzes — all free.
Start learning freeContinue reading
- Probability | Class 10 Mathematics Notes
Clear NCERT-aligned notes on Probability for Class 10 Mathematics.
- Probability | Class 10 Mathematics Notes
Clear NCERT-aligned notes on Probability for Class 10 Mathematics.
- Probability | Class 10 Mathematics Notes
Clear NCERT-aligned notes on Probability for Class 10 Mathematics.