What is the difference between a tangent and a secant?

A secant intersects a circle at two distinct points. A tangent touches the circle at exactly one point — it is the limiting case of a secant as its two intersection points merge into one.

What is the relationship between a tangent and the radius?

The tangent at any point of a circle is always perpendicular to the radius drawn to that point of contact (Theorem 10.1). This is because the radius is the shortest distance from the centre to any point on the tangent line.

How many tangents can be drawn from a point to a circle?

If the point is inside the circle, zero tangents can be drawn. If the point lies on the circle, exactly one tangent exists. If the point is outside the circle, exactly two tangents can be drawn.

Are the two tangent lengths from an external point always equal?

Yes (Theorem 10.2). The two tangent segments drawn from the same external point to a circle are always equal in length, proved using RHS congruence of the two right triangles formed with the centre.

What is the first step in solving any tangent problem?

Draw the radius from the centre to each point of contact. This immediately creates a right angle (tangent is perpendicular to radius), which is almost always the key to setting up the rest of the solution.

Chapter 10 · CBSE Class 10 Maths

Circles

Two ways to learn — a crisp formula reference, or a story that makes the tangent theorems feel obvious.

Tangent vs secantTangent ⊥ radiusNumber of tangentsEqual tangent lengths

Also useful:Chapter 9 — Some Applications of Trigonometry Chapter 11 — Areas Related to Circles NCERT Solutions — Class 10 Maths 1-on-1 CBSE Class 10 Tuition

CBSEClass 10MathematicsChapter 10⏱ 7 min crisp · 10 min story

Four scenes that build the ideas from scratch, step by step.

How this works

Four scenes, one thread: a tangent is just a secant whose two crossing points have merged into one — and that single idea, plus one right angle, unlocks everything else in this short chapter. Read the story, then tap "The Maths" for the formal result.

Three Ways a Line Meets a Circle

Watch a bicycle wheel roll along the ground. At any instant, the ground touches the wheel at exactly ONE point — never two, never zero (while it's rolling on the ground, at least). That single contact point is what a tangent is.

Student

So a tangent is just a special, extreme case of a line crossing a circle at two points?

Exactly. Slide a two-point secant line sideways, and its two intersection points drift closer and closer together. Right at the moment they merge into one, the secant has become a tangent.

Line misses the circle entirely → 0 intersection points (non-intersecting)

Line crosses the circle → 2 intersection points (secant)

Line just grazes the circle → 1 intersection point (tangent) — the limiting case

Now watch the wheel again, this time tracking its spoke (the radius) to the ground-contact point. That spoke always looks exactly perpendicular to the ground, in every single position as the wheel rolls.

Inside, On, or Outside

Try drawing a tangent through a point strictly INSIDE a circle. No matter how the line is angled, it always crosses the circle at two points — never manages to graze it at just one.

Teacher

Now try a point ON the circle itself, then a point OUTSIDE it. How many tangents can be drawn from each?

Point INSIDE the circle → 0 tangents possible (every line through it is a secant)

Point ON the circle → exactly 1 tangent (the one from Scene 1)

Point OUTSIDE the circle → exactly 2 tangents

The outside case is the interesting one: from an external point P, two distinct lines PT₁ and PT₂ can both be drawn touching the circle, at two different points of contact T₁ and T₂.

A very common gap: forgetting there are always TWO tangents from an outside point, and only working with one of them — missing the symmetry (equal lengths, angle bisector) that comes from having both.

Equal Tangent Lengths

From external point P, two tangents touch the circle at Q and R. Measure PQ and PR — remarkably, they always come out exactly equal, no matter where P is.

Teacher

Draw OP, OQ, and OR. What do you notice about triangles OQP and ORP?

∠OQP = ∠ORP = 90° (tangent ⊥ radius, Theorem 10.1, at BOTH points of contact)

OQ = OR (both radii of the same circle)

OP = OP (shared side)

△OQP ≅ △ORP (RHS congruence) → PQ = PR

A neat consequence falls out for free: since OP is shared and the triangles are congruent, ∠OPQ = ∠OPR too — meaning OP bisects the angle between the two tangents.

Real application: two concentric circles, radii 5 cm and 3 cm — find the chord of the bigger circle that's tangent to the smaller one. The radius to the tangent point is perpendicular to the chord (Theorem 10.1) AND bisects it (perpendicular from centre bisects a chord) — giving half-chord = √(5²−3²) = 4, so the full chord is 8 cm.

The Isosceles Triangle Trick

Since PQ = PR always (Scene 3), triangle PQR — formed by an external point and its two points of contact — is automatically isosceles, every single time. That fact alone unlocks angle relationships without extra work.

Student

If ∠PTQ = θ (the angle between two tangents from point T), what's ∠OPQ in terms of θ?

△TPQ is isosceles (TP=TQ) → ∠TPQ = ∠TQP = (180°−θ)/2 = 90° − θ/2

∠OPT = 90° (tangent ⊥ radius)

∠OPQ = ∠OPT − ∠TPQ = 90° − (90° − θ/2) = θ/2

So ∠PTQ = 2×∠OPQ always — a direct, reusable relationship, not something to re-derive each time.

A different real problem: chord PQ = 8 cm in a circle of radius 5 cm, tangents at P and Q meet at T. Finding TP needs a slightly different tool — similar triangles.

OT bisects PQ (Scene 3's angle-bisector fact) at R, so PR = 4, OR = √(5²−4²) = 3

∠RPO = ∠PTR (both complementary to the same angle) → △TRP ~ △PRO (AA similarity)

TP/PO = RP/RO → TP/5 = 4/3 → TP = 20/3 cm

The Big Picture

Every idea in this chapter branches from one picture: a tangent is a secant whose two points have merged into one, and it's always perpendicular to the radius at that point. Combine that right angle with the equal-tangent-lengths fact, and nearly every problem in this chapter falls into place.

What is a tangent, really?

A secant with its 2 points merged into 1

Angle between tangent and radius?

Always 90° (Theorem 10.1)

Tangents from inside/on/outside?

0 / 1 / 2 respectively

Two tangents from one point — lengths?

Always equal (Theorem 10.2)

First move in any tangent problem?

Draw the radius to the point of contact

Want to go deeper?

Our tutors cover the full CBSE Class 10 Maths syllabus live, 1-on-1. If Chapter 10 still feels shaky after this, one session usually fixes it.

Book a Free Trial Class