Chapter 13 of 19

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Andhra Pradesh (BIEAP)Class 10 Mathematics★ AP Board (BSEAP) SSC Mathematics · ~4 marks · Board: AP Board

Tangents and Secants to a Circle

TANGENTS (touch at one point), SECANTS (intersect at two points). Tangent is PERPENDICULAR to radius. Tangents from an external point are EQUAL. AP Board Class 10 Mathematics.

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By the end of this chapter you'll be able to…

1Distinguish between tangent and secant to a circle
2State and apply: tangent is perpendicular to the radius at the point of contact
3Prove and apply: tangents from an external point are equal in length
4Calculate the length of a tangent from an external point given the distance from centre and radius
5Apply the Alternate Segment Theorem in angle-finding problems

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Why this chapter matters

Tangents and Secants is a circle geometry chapter that tests specific theorems and calculation skills. The two key theorems — tangent perpendicular to radius, and equal tangents from external point — are tested in both proof and application forms. The length-of-tangent formula is used in every numerical problem in this chapter. This is a compact chapter that yields 4 reliable marks with minimal study time.

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Tangents and Secants to a Circle

Definitions

Tangent: Line touching the circle at EXACTLY ONE point. Secant: Line intersecting the circle at TWO points.

Key Theorems

Tangent is PERPENDICULAR to the radius at the point of contact.
Tangents from an EXTERNAL point are EQUAL in length.
Alternate Segment Theorem: Angle between tangent and chord = angle in ALTERNATE segment.

Length of Tangent: From external point P to circle with centre O and radius r: PT = √(OP² − r²).

Common Mistakes: Assuming secant and tangent are the same. 'Tangent touches ONCE. Secant cuts TWICE.'

Secant-Tangent Theorem (Tangent-Secant Power Theorem)

Statement: If a TANGENT and a SECANT intersect at an external point, the SQUARE of the length of the tangent segment equals the PRODUCT of the secant segment and its external segment.

If PT is the tangent and PAB is the secant from point P: PT² = PA × PB

'Where PA is the EXTERNAL part (outside the circle) and PB is the FULL secant (PA + AB).'

Proof

Consider circle with centre O. Tangent PT touches at T. Secant PAB cuts the circle at A and B. Join T to A and B. In △PTA and △PBT: ∠PTA = ∠PBT (Alternate Segment Theorem — angle between tangent and chord equals angle in alternate segment) ∠P = ∠P (common) △PTA ∼ △PBT (AA criterion) Therefore, PT/PB = PA/PT Cross-multiplying: PT² = PA × PB ✓

Example 1: From an external point P, a tangent PT of length 6 cm is drawn to a circle. If the secant PAB has external part PA = 4 cm, find the length of the full secant PB.

PT² = PA × PB 6² = 4 × PB → 36 = 4PB → PB = 9 cm ✓ 'So the chord AB = PB − PA = 9 − 4 = 5 cm.'

Secant-Secant Theorem (Two-Secant Power Theorem)

Statement: If TWO SECANTS intersect at an external point, the PRODUCT of one secant and its external part equals the PRODUCT of the other secant and its external part.

If PAB and PCD are two secants from point P: PA × PB = PC × PD

Example 2: Two secants PAB and PCD intersect at external point P. If PA = 3 cm, PB = 12 cm, and PC = 4 cm, find PD.

PA × PB = PC × PD 3 × 12 = 4 × PD → 36 = 4PD → PD = 9 cm ✓

Example 3: In the same setup, if PA = 5, AB = 7, and PC = 4, find CD. PB = PA + AB = 5 + 7 = 12 PA × PB = PC × PD → 5 × 12 = 4 × PD → PD = 15 CD = PD − PC = 15 − 4 = 11 cm ✓

'Secant-Secant theorem is especially useful in construction problems and proof-based questions in the AP Board exam.'

Tangent From an External Point — Deep Dive

Theorem: Tangents drawn from an EXTERNAL point to a circle are EQUAL in length.

Given: External point P. Tangents PT and PS to circle with centre O. To Prove: PT = PS

Proof: Join O to P, T, and S. OT ⟂ PT (tangent is perpendicular to radius at point of contact) OS ⟂ PS (same reason) In right △OTP and △OSP: OT = OS (radii of same circle) OP = OP (common side) △OTP ≅ △OSP (RHS congruence) Therefore, PT = PS ✓ (CPCT)

Example 4: From point P outside a circle of radius 5 cm, two tangents PT and PS are drawn. If OP = 13 cm, find PT and the angle between the tangents.

PT = √(OP² − r²) = √(169 − 25) = √144 = 12 cm PS = PT = 12 cm (tangents from same external point are EQUAL) In △OTP: sin(∠OPT) = OT/OP = 5/13 ∠OPT = arcsin(5/13) ≈ 22.6° ∠TPS = 2 × ∠OPT ≈ 45.2° ✓

Angle Between Tangent and Chord (Alternate Segment Theorem)

Statement: The angle between a TANGENT and a CHORD drawn at the point of contact is EQUAL to the angle in the ALTERNATE SEGMENT.

If ∠PTA is the angle between tangent PT and chord TA, then ∠PTA = ∠TBA where B is any point on the circle in the alternate (opposite) segment.

'Think of the chord dividing the circle into TWO segments. The angle on one side equals the angle between the chord and tangent on the OTHER side.'

Example 5: In a circle, the tangent at A makes an angle of 60° with chord AB. Find the angle in the alternate segment.

By Alternate Segment Theorem: The angle in the alternate segment = 60° ✓

Construction of Tangents to a Circle

Case 1: Tangent at a Point ON the Circle

Draw the radius OA to the given point A.
Construct a line ⟂ OA at A.
This line is the REQUIRED tangent.

'There is EXACTLY ONE tangent at any point on the circle.'

Case 2: Tangents from a Point OUTSIDE the Circle

Step 1: Join the external point P to the centre O. Find the MIDPOINT M of OP. Step 2: With M as centre and MO as radius, draw a circle that cuts the given circle at TWO points (call them T and S). Step 3: Join PT and PS. These are the REQUIRED tangents.

'From an external point, EXACTLY TWO tangents can be drawn. They are EQUAL in length.'

Example 6: Construct a tangent to a circle of radius 3 cm from a point 7 cm from the centre. Length of tangent = √(OP² − r²) = √(49 − 9) = √40 ≈ 6.32 cm ✓

Angle Between Two Tangents

If two tangents are drawn from an external point P to a circle, the angle between them is: sin(θ/2) = r/OP

Example 7: Two tangents are drawn from point P to a circle of radius 4 cm. If OP = 8 cm, find the angle between the tangents.

sin(θ/2) = r/OP = 4/8 = 1/2 θ/2 = 30° θ = 60° ✓

Common Mistakes — Expanded

Mistake	Correction
Thinking tangent = secant	Tangent touches ONCE. Secant cuts TWICE. NEVER the same.
Forgetting to square PT in PT² = PA×PB	It is PT SQUARED, not PT. Many students forget the square.
Wrong application of alternate segment theorem	The angle is between TANGENT and CHORD — not between two chords.
Tangents from a point inside the circle	Tangents exist ONLY from points OUTSIDE or ON the circle.
Using chord-chord power theorem with secants	Chord-chord (inside circle) is different from secant-secant (outside).

Applications in Real Life

Reflection in circular mirrors: Tangent properties describe how light reflects off curved surfaces.
Satellite dishes: The reflector shape uses tangent properties for signal focusing.
Gear design: Tangents to pitch circles determine where gear teeth make contact.
Track design: Circular race tracks have tangent straight segments for entry and exit.

AP SSC Board Exam Focus

Topic	Marks	Frequency
Tangent ⟂ radius theorem	2-3	Very High
Tangents from external point (equal lengths)	3-4	Very High
Secant-tangent theorem (PT² = PA×PB)	4	High
Alternate segment theorem	3-4	High
Construction of tangents	4	Moderate
Secant-secant theorem	3	Moderate
Length of tangent calculation	2-3	Very High

Self-Test Questions

A tangent PT is drawn from point P to a circle of radius 5 cm. If OP = 13 cm, find PT.
From an external point P, a secant PAB cuts the circle at A and B. If PA = 4 cm and PB = 9 cm, find the length of tangent PT from P.
Two secants PAB and PCD are drawn from point P. If PA = 5 cm, AB = 7 cm, PC = 6 cm, find CD.
Construct two tangents to a circle of radius 4 cm from a point 9 cm from the centre. Measure and verify their lengths.
In a circle, the tangent at point A makes an angle of 45° with chord AB. What is the angle in the alternate segment?
Two tangents are drawn from point P to a circle. If the distance from P to the centre is twice the radius, find the angle between the tangents.
Prove that the tangents drawn at the ends of a DIAMETER of a circle are PARALLEL.
A circle has radius 6 cm. From a point 10 cm from the centre, a tangent is drawn. Find its length.

Answers: 1) 12 cm, 2) 6 cm, 3) 4 cm, 4) Construction steps as above (length ≈ 8.06 cm), 5) 45°, 6) 60°, 7) Both tangents are ⟂ to the same diameter → parallel to each other, 8) 8 cm

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Tangent Theorems

THEOREM 1: Tangent ⊥ Radius at point of contact. If PT is tangent at T, O is centre, then OT ⊥ PT. THEOREM 2: Tangents from external point are EQUAL. If PA and PB are tangents from P to circle with centre O, then PA = PB. Corollary: PO bisects angle APB and angle AOB. LENGTH OF TANGENT: If P is external point, O is centre, r = radius, OP = distance. Tangent length = PT = √(OP² − r²) [by Pythagoras in right ΔOTP].

PROOF OF EQUAL TANGENTS (AP Board asks this): Consider tangents PA and PB from external point P. Join OA, OB, OP. In ΔOAP and ΔOBP: OA = OB (radii). OP = OP (common). ∠OAP = ∠OBP = 90° (tangent ⊥ radius). By RHS: ΔOAP ≅ ΔOBP. Therefore PA = PB.

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Confusing secant and tangent, especially in theorems

✓ TANGENT: Touches the circle at EXACTLY ONE POINT. Creates a RIGHT ANGLE with the radius at that point. SECANT: Passes through the circle, cutting it at TWO POINTS. Remember: 'Tangent = touch (T).' In diagrams, always draw the radius to the point of tangency first — this creates the right angle you need for Pythagoras.

Practice problems

Work through this chapter's problems as a readiness check — reveal each solution, mark yourself honestly, and get your gap report at the end.

Readiness check

Are you exam-ready for Tangents and Secants to a Circle?

1 problems from this chapter. Try each one, reveal the worked solution, mark yourself honestly — get your gap report at the end.

1 questions~2 min

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•TANGENT: A line that touches a circle at EXACTLY ONE POINT (the point of tangency/contact). SECANT: A line that intersects a circle at TWO POINTS (passes through the circle). A chord is the segment of a secant between the two intersection points.
•THEOREM 1 — TANGENT ⊥ RADIUS: At the point of tangency, the tangent is PERPENDICULAR to the radius. If T is the point of tangency, O is the centre, then OT ⊥ PT (where P is any other point on the tangent). This creates a RIGHT ANGLE at T — key for applying Pythagoras.
•THEOREM 2 — EQUAL TANGENTS FROM EXTERNAL POINT: If PA and PB are two tangents drawn from external point P to a circle, then PA = PB. The tangent lengths from any external point are equal. Corollary: PO bisects ∠APB (the angle between the two tangents) and ∠AOB.
•TANGENT LENGTH FORMULA: In right triangle OTP (right angle at T): PT² = OP² − OT² = OP² − r². So tangent length PT = √(OP² − r²). Here OP = distance from external point to centre, r = radius. This is direct Pythagoras in the right triangle formed by radius, tangent, and line to centre.
•PROOF OF EQUAL TANGENTS: Join OA, OB, OP. In ΔOAP and ΔOBP: OA = OB (radii, equal). OP = OP (common side). ∠OAP = ∠OBP = 90° (tangent perpendicular to radius). By RHS congruence: ΔOAP ≅ ΔOBP. Therefore PA = PB. This proof is frequently asked in AP SSC.
•ANGLE BETWEEN TANGENTS: If ∠APB = x° (angle between two tangents from P), then ∠AOB = 180° − x°. This is because quadrilateral AOBP has ∠OAP + ∠OBP = 90° + 90° = 180°, so ∠APB + ∠AOB = 360° − 180° = 180°. Therefore ∠AOB = 180° − ∠APB.
•ALTERNATE SEGMENT THEOREM (Tangent-Chord Angle): The angle between a tangent to a circle and a chord drawn from the point of tangency equals the inscribed angle on the opposite side of the chord. Used for angle-chasing problems in AP SSC.
•SECANT-TANGENT RELATIONSHIP: If a tangent PT and a secant PAB are drawn from external point P: PT² = PA × PB (tangent-secant theorem). This is tested as a formula application in some AP SSC papers.
•NUMBER OF TANGENTS FROM A POINT: From a point OUTSIDE the circle: exactly 2 tangents. From a point ON the circle: exactly 1 tangent. From a point INSIDE the circle: 0 tangents (no tangent can be drawn from inside). This classification is tested as a 1-mark concept question.
•COMMON TANGENTS: Two circles can have 0, 1, 2, 3, or 4 common tangents depending on their positions. External common tangents: 2 (circles don't overlap). Internal common tangents: 2 (when circles don't touch). When circles are externally tangent: 3 common tangents. When circles overlap: 2 external only. When one is inside the other: 0.

Andhra Pradesh (BIEAP) marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Pulley and belt systems in machines

In mechanical engineering, a belt running between two circular pulleys is tangent to both circles. The tangent length between the pulleys determines the belt length and the angle of 'wrap' (how much of each pulley the belt contacts). Engineers use the equal tangent theorem and the tangent-length formula constantly when designing conveyor belts, timing belts in car engines, and industrial drive systems. The same geometry applies to bicycle chains and gears.

Satellite orbit tangent manoeuvres

When a satellite needs to change from one circular orbit to another (e.g., moving from low Earth orbit to geostationary orbit), it uses a Hohmann transfer orbit — an elliptical path tangent to both circular orbits. The tangent condition (touching both circles at exactly one point) ensures the smooth transition. Understanding tangent lines to circles is the geometric foundation of orbital mechanics. ISRO mission planners use this principle for every satellite launch.

Road design and circular curves

When a straight road meets a circular curve (like a roundabout or highway bend), the transition must be tangential — the straight road must be a tangent to the circular curve at the joining point. This ensures vehicles don't experience a sudden change in direction. Civil engineers design these tangent transitions carefully: the 'tangent point' is the point where the straight road becomes the circular curve — exactly the point of tangency in our chapter. Highway design specifications list tangent lengths explicitly.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Tangent length calculation (2 marks): identify OP (centre to external point), r (radius). Apply PT = √(OP²−r²). Write Pythagoras explicitly: PT² = OP² − r². Show subtraction, then square root. Two steps clearly shown = 2 marks.
Equal tangents proof (4 marks): the RHS congruence proof has FOUR components — state OA=OB (radii), OP=OP (common), ∠OAP=∠OBP=90° (tangent⊥radius), then conclude ΔOAP≅ΔOBP by RHS, therefore PA=PB. Each component earns a mark — don't skip any.
Angle between tangents (2 marks): write ∠APB + ∠AOB = 180°. Substitute the given angle. Calculate the other. Cite that AOBP is a quadrilateral with two right angles (at A and B) — this justification earns the full marks.
Drawing: always draw a clear circle, mark the centre O, mark the external point P, draw both tangents with their tangent points A and B clearly labelled. The diagram helps you identify which triangles to use.
Alternative segment theorem questions: identify the tangent, the chord, and the inscribed angle in the alternate segment. State the theorem explicitly: 'By Alternate Segment Theorem, ∠ between tangent and chord = ∠ inscribed in alternate segment.' Then write the equal angles.

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Research the Nine-Point Circle and its tangent properties — a remarkable circle that passes through nine specific points of any triangle: the midpoints of the three sides, the feet of the three altitudes, and the midpoints of the three segments from each vertex to the orthocentre. The nine-point circle is internally tangent to the incircle and externally tangent to the three excircles — a beautiful theorem by Feuerbach (1822). Research the relationships between the incircle, excircles, and nine-point circle.
Investigate inversive geometry — a transformation that maps each point P to a point P' such that OP × OP' = r² (where O is the centre of inversion circle, r is the radius). Under inversion, circles through O map to lines, circles not through O map to circles. The tangent-secant theorem (PT² = PA × PB) is precisely the inversion formula: T is the inverse of A (and of B) with respect to the circle. Research how inversion elegantly proves many classical circle theorems.
Explore the Pappus chain — if two circles of different radii are mutually tangent and both internally tangent to a large circle, a chain of smaller circles can be placed in the 'arbelos' region, each tangent to its neighbours and to the two original circles. Remarkably, the height of the nth circle in the chain above the base line equals n times the diameter of the nth circle — a theorem dating to Pappus of Alexandria (c. 300 CE), now proved elegantly using inversive geometry.
Research the common external and internal tangents to two circles in engineering — when designing belt-and-pulley systems, gear trains, or lens systems in optics, engineers compute the length of common external tangents. The formula: for two circles with centres distance d apart and radii r₁ and r₂: external tangent length = √(d²−(r₁−r₂)²), internal tangent length = √(d²−(r₁+r₂)²). Research how these formulas are derived from the Pythagorean theorem.

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

AP Board SSC (Class 10)Medium — Tangents to circles appear for 2–4 marks; the tangent-length calculation and equal tangents proof are the most common question types

JEE Main (Coordinate Geometry — Circles)High — Tangent to a circle (equation of tangent, length of tangent, common tangents to two circles) is a standard JEE topic; Class 10 is the conceptual foundation

NTSE (Mathematics)Medium — Tangent properties and circle theorems appear in NTSE Stage I and II

AP EAPCET (Mathematics)High — Circle geometry (equations, tangents, chords) is a major chapter in EAPCET Mathematics

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Questions students ask

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PROOF BY CONTRADICTION: Suppose the radius OT is NOT perpendicular to the tangent at T. Then there exists a point T' on the tangent where OT' ⊥ tangent (drop perpendicular from O to the tangent line). In right triangle OTT': OT' < OT (the perpendicular from a point to a line is the shortest distance). But OT = radius, and T is on the circle. Since OT' < OT = r, the point T' is INSIDE the circle. But T' is on the tangent — which should only touch the circle at one point (not pass through it). This contradiction means OT MUST be the shortest distance, so OT is perpendicular to the tangent. The shortest distance from a point to a line is always the perpendicular.

TANGENT-SECANT THEOREM: If PT is a tangent from external point P, and PAB is a secant through P intersecting the circle at A and B (A closer to P), then PT² = PA × PB. This is also written as PT² = PA × PB. This means the tangent length squared = product of the two distances to the secant intersection points. Example: P is 5 cm from centre, radius = 3 cm. Tangent length PT = √(25−9) = 4 cm. If a secant from P has PA = 2, then PB = PT²/PA = 16/2 = 8 cm. This theorem connects tangent and secant lengths elegantly.

Use the fact that AOBP is a quadrilateral with ∠OAP = ∠OBP = 90°. Sum of angles in quadrilateral = 360°. So ∠APB + ∠AOB + 90° + 90° = 360°. Therefore ∠APB + ∠AOB = 180° (supplementary). If ∠APB = 70°, then ∠AOB = 180° − 70° = 110°. ALTERNATIVELY: If ∠APB = 40°, then ∠AOB = 140°. When the tangents are equal in length and ∠APB is given, you can also find the tangent length if OP is given — by using the right triangle OAP with ∠APO = ∠APB/2 = 20° (since PO bisects ∠APB).

Both touch but don't cross — but they differ fundamentally: A TANGENT TO A CIRCLE touches the circle at exactly one point and then moves away. At the point of contact, the circle and tangent have the same 'direction' (the tangent IS the limiting secant as the two intersection points merge). AN ASYMPTOTE to a curve (like a hyperbola) approaches the curve infinitely but never actually touches it — the distance between them approaches zero but remains positive. A tangent to a circle DOES touch (distance = 0 at the point of contact). In calculus (Class 11), the tangent to any curve at a point is the line whose slope equals the derivative at that point — the circle tangent is a special case where the radius provides the perpendicular.

OP = distance from O(0,0) to P(13,0) = 13 cm. Radius r = 5 cm. Tangent length PT = √(OP²−r²) = √(169−25) = √144 = 12 cm. Verify using the Pythagorean triple (5,12,13): the right triangle OTP has OT=5, PT=12, OP=13 — a perfect Pythagorean triple ✓. The right angle is at T (tangent ⊥ radius).

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