Schooling · Class 10 · Mathematics · Circles · Chapter on a page
CBSE Class 10 · Mathematics · NCERT Mathematics
Circles: chapter on a page
A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).
About the notes: Written by Shishya's AI on 26 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.
This note's current text has no applied fact-check on record.
Mind map
Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.
The mind map as a list
- Key ideas
- A tangent to a circle is a line that…
- The tangent at any point on a circle…
- From a point outside a circle…
- The lengths of the two tangent…
- The line joining an external point…
- If two tangents are drawn from an…
- The perpendicular from the centre of…
- Formulas and facts
- Tangent perpendicular to radius
- Equal tangent lengths
- Angle between tangents
- Right triangle formed
- Watch out for
- Forgetting that tangent meets radius…
- Assuming tangent passes through the…
- Using the wrong triangle for…
- Thinking unequal tangent lengths…
- Confusing tangent with chord or secant
What this chapter is about
This chapter explores the relationship between circles and lines, particularly tangents. You already know basic circle concepts from earlier classes — radius, diameter, chord, arc, sector. Now you study how a line can touch a circle at exactly one point and what special properties this creates.
Key ideas
- A tangent to a circle is a line that touches the circle at exactly one point; a secant is a line that cuts the circle at two points.
- The tangent at any point on a circle is perpendicular to the radius drawn to that point of contact.
- From a point outside a circle, exactly two tangents can be drawn to the circle.
- The lengths of the two tangent segments drawn from an external point to a circle are equal.
- The line joining an external point to the centre of the circle bisects the angle between the two tangents from that point.
- If two tangents are drawn from an external point, the line joining the external point to the centre passes through the midpoint of the chord joining the two points of contact.
- The perpendicular from the centre of a circle to a chord bisects the chord (useful when tangents and chords appear together in problems).
Formulas and facts to remember
- Tangent perpendicular to radius: If PT is a tangent to a circle with centre O at point T, then OT ⊥ PT.Meaning: The radius and tangent meet at a right angle.
- Equal tangent lengths: If PA and PB are tangents from external point P to a circle with centre O, touching at A and B, then PA = PB.Meaning: Both tangent segments from the same outside point have the same length.
- Angle between tangents: If PA and PB are tangents from P, then ∠APB and ∠AOB are supplementary, meaning ∠APB + ∠AOB = 180°.Meaning: The angle at the external point and the angle at the centre (formed by the two radii to the contact points) add up to 180°.
- Right triangle formed: Triangle OAP (where O is centre, A is contact point, P is external point) is right-angled at A.Meaning: You can apply Pythagoras theorem: OP² = OA² + PA².
Common mistakes
- Forgetting that tangent meets radius at 90° → Always draw the radius to the contact point and mark the right angle before solving.
- Assuming tangent passes through the centre → The tangent never passes through the centre; it only touches the circle at one point on the circumference.
- Using the wrong triangle for Pythagoras → The right angle is at the point of contact, not at the external point or the centre.
- Thinking unequal tangent lengths from an external point → Both tangents from the same external point are always equal in length.
- Confusing tangent with chord or secant → A tangent touches at one point only; a chord joins two points on the circle; a secant cuts through two points.
Quick revision
- Tangent touches circle at exactly one point; perpendicular to radius there.
- From an external point, two tangents can be drawn, both of equal length.
- Use Pythagoras in the right triangle formed by radius, tangent and line to centre.
- ∠APB + ∠AOB = 180° when PA and PB are tangents from external point P.
- The radius to the contact point is your key line — draw it first in every problem.