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CBSE · Class 10 · Mathematics

Circles 10.2: Tangent to a Circle

5 concepts · 40 practice questions with answers · Free to read

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Secant, tangent and point of contact

From daily life

A cycle wheel touches the road at just one spot, while a plank sliding across a pond cuts the water twice. A line and a circle meet in the same few ways.

Simple definition

A line meeting a circle at two points is a secant. A line touching it at exactly one point is a tangent, and that point is the point of contact. A line may also miss the circle. A tangent never enters the inside of the circle.

OTsecant: 2 pointstangent: 1 pointmisses: 0 points

In the figure: Two common points: secant. One: tangent. None: the line misses the circle.

Rule 1: Count the common points — The number of common points names the line.

  • Easy: Two common points: the line is a secant.
  • Medium: One common point: the line is a tangent.
  • Exam-type: Through a point on the circle exactly one tangent can be drawn.

Rule 2: Test with the distance from the centre — Compare the distance d of the line from the centre with the radius r.

  • Easy: r = 5 cm, d = 3 cm: d < r, so the line is a secant.
  • Medium: r = 5 cm, d = 5 cm: d = r, so the line is a tangent.
  • Exam-type: r = 5 cm, d = 7 cm: d > r, so the line misses the circle.
Remember

A tangent touches the circle at one point only; it does not cut across it.

Common mistake

Calling a line that just looks close to the circle a tangent; it is a tangent only if it has exactly one common point.

Memory clue

"Two points cut, one point touches, none misses."

Practice questions

Easy
  1. 1The rope on either side of a pulley fitted over a well is like a ___ to the circular pulley.
  2. 2A circle has radius 26 cm and a line is 26 cm from its centre. How many points do the line and the circle have in common?
  3. 3A tangent to a circle meets it in how many points?
Medium
  1. 4A line is 46 cm from the centre of a circle of radius 40 cm. By how many cm must it be moved towards the centre so that it just touches the circle?
  2. 5A line is 18 cm from the centre of a circle of radius 40 cm. By how many cm must it be moved directly away from the centre so that it just touches the circle?
  3. 6A line is 2 cm from the centre of a circle of radius 34 cm. By how many cm must it be moved directly away from the centre so that it just touches the circle?
Challenging
  1. 7Three lines l1, l2, l3 are at distances 33 cm, 7 cm and 22 cm from the centre of a circle of radius 22 cm. How many of them are tangents?
  2. 8Three lines l1, l2, l3 are at distances 15 cm, 3 cm and 2 cm from the centre of a circle of radius 3 cm. How many of them are tangents?
Show answers (try them yourself first)
  1. 1tangent
  2. 21
  3. 31
  4. 46
  5. 522
  6. 632
  7. 71
  8. 81

Tangent is perpendicular to the radius

From daily life

A stone whirled on a string flies off along the straight line it was moving on, at right angles to the string, when you let go.

Simple definition

The tangent at a point on a circle is perpendicular to the radius through the point of contact: OT ⟂ PT, so ∠OTP = 90°. Conversely, a line perpendicular to a radius at its outer end is a tangent.

Formula: OT ⟂ PT; ∠OTP = 90°

OTPOT ⟂ PTangle OTP = 90°r

In the figure: The radius OT meets the tangent at T at a right angle.

Rule 1: Use the right angle — Triangle OTP has a right angle at T, so the other two angles add to 90°.

  • Easy: ∠TOP = 50°: ∠TPO = 90° − 50° = 40°.
  • Medium: ∠TPO = 25°: ∠TOP = 90° − 25° = 65°.
  • Exam-type: The angle between the tangent and the radius is always 90°.

Rule 2: Perpendicular through the contact point — The perpendicular to a tangent at its point of contact passes through the centre.

  • Easy: To find the centre, draw perpendiculars at two tangent points; they meet at O.
  • Medium: A line perpendicular to a radius at its end on the circle is a tangent.
  • Exam-type: At that point no other tangent exists.
Remember

Whenever a tangent appears, draw the radius to the point of contact and mark 90°.

Common mistake

Joining the point of contact to some other point instead of the centre; the right angle is only with the radius.

Memory clue

"Tangent and radius meet at a right angle."

Practice questions

Easy
  1. 1The tangent at V to a circle with centre O meets a line through O at T. In the triangle OVT, ∠OTV = 35°. Find ∠VOT (in degrees).
  2. 2KS is a tangent to a circle with centre O at the point K. If ∠KOS = 50°, find ∠OSK (in degrees).
  3. 3The tangent at D to a circle with centre O meets a line through O at S. In the triangle ODS, ∠OSD = 60°. Find ∠DOS (in degrees).
Medium
  1. 4The tangent at N to a circle with centre O meets a line through O at S. In the triangle ONS, ∠OSN = 30°. Find ∠NOS (in degrees).
  2. 5BS is a chord of a circle with centre O and ∠BOS = 100°. The tangent at B is BE. Find ∠EBS (in degrees).
  3. 6ZX is a chord of a circle with centre O and ∠ZOX = 20°. The tangent at Z is ZA. Find ∠AZX (in degrees).
Challenging
  1. 7BM is a chord of a circle with centre O and ∠BOM = 50°. The tangent at B is BV. Find ∠VBM (in degrees).
  2. 8FD touches a circle with centre O at F. In the triangle OFD, ∠FOD : ∠ODF = 2 : 7. Find the larger acute angle (in degrees).
Show answers (try them yourself first)
  1. 155
  2. 240
  3. 330
  4. 460
  5. 550
  6. 610
  7. 725
  8. 870

Length of a tangent (Pythagoras)

From daily life

To know how far a lamp post P stands from where a straight path touches a round pond, you need no tape along the path.

Simple definition

In the right triangle OTP, OP is the hypotenuse. So the length of the tangent from P is PT = √(OP² − r²). Know any two of OP, r and PT and the third follows by Pythagoras.

Formula: PT² = OP² − r²

OTP5PT = ?OP = 13PT² = OP² − r²= 169 − 25= 144, PT = 12

In the figure: r = 5, OP = 13: PT = √(169 − 25) = √144 = 12.

Live figure

Length of a tangent (Pythagoras)

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Rule 1: Find PT — Square OP and r, subtract, take the root.

  • Easy: r = 5 cm, OP = 13 cm: PT = 12 cm.
  • Medium: r = 6 cm, OP = 10 cm: PT = √(100 − 36) = 8 cm.
  • Exam-type: r = 7 cm, OP = 25 cm: PT = √(625 − 49) = 24 cm.

Rule 2: Find OP or r — Rearrange the same relation.

  • Easy: PT = 15 cm, r = 8 cm: OP = √(225 + 64) = 17 cm.
  • Medium: PT = 24 cm, OP = 25 cm: r = √(625 − 576) = 7 cm.
  • Exam-type: PT is always shorter than OP.
Remember

The tangent length from P is the same whichever side of the circle you take; there are two of them.

Common mistake

Adding the squares instead of subtracting; OP is the hypotenuse, so PT² = OP² − r².

Memory clue

"Big square minus radius square, then root."

Practice questions

Easy
  1. 1A tangent SQ at a point S of a circle with centre O has length 25 cm and the radius is 60 cm. Find OQ (in cm).
  2. 2A tangent CG at a point C of a circle with centre O has length 25 cm and the radius is 60 cm. Find OG (in cm).
  3. 3From a point V the length of the tangent to a circle is 12 cm and the distance of V from the centre is 13 cm. Find the radius of the circle (in cm).
Medium
  1. 4A tangent YW to a circle with centre O touches it at Y. If the radius is 3 cm and YW = 4 cm, find the area of the triangle OYW (in cm²).
  2. 5A tangent AP to a circle with centre O touches it at A. If the radius is 15 cm and AP = 20 cm, find the area of the triangle OAP (in cm²).
  3. 6The radius of a circle is 5 cm and a point M is 14 cm from its centre. Find the length of the tangent MK from M to the circle (in cm).
Challenging
  1. 7From a point L the distance to the centre of a circle of radius 16 cm exceeds the length of the tangent from L by 4 cm. Find the length of the tangent (in cm).
  2. 8The radius of a circle with centre O is 18 cm. The tangent UK at U has length 24 cm. Find the perimeter of the triangle OUK (in cm).
Show answers (try them yourself first)
  1. 165
  2. 265
  3. 35
  4. 46
  5. 5150
  6. 63√19
  7. 730
  8. 872

Concentric circles and chords

From daily life

On a dartboard the rings share one centre. A straight line just touching an inner ring cuts the outer ring into a chord.

Simple definition

Two circles with the same centre are concentric. A chord AB of the larger circle (radius R) that touches the smaller circle (radius r) at T is bisected at T, and AT² = R² − r², so AB = 2√(R² − r²).

Formula: AB = 2√(R² − r²)

OTAB44R = 5, r = 3AT² = R² − r²= 25 − 9 = 16AB = 2 × 4 = 8

In the figure: R = 5, r = 3: AT = √(25 − 9) = 4, so the chord AB = 8.

Rule 1: Find the chord — Half the chord is the other leg of the right triangle.

  • Easy: R = 5 cm, r = 3 cm: chord = 2 × 4 = 8 cm.
  • Medium: R = 13 cm, r = 5 cm: chord = 2 × 12 = 24 cm.
  • Exam-type: The point of contact T is the mid-point of the chord.

Rule 2: Find a radius — Use the half chord and the other radius.

  • Easy: Chord 16 cm, R = 10 cm: half chord 8, so r = √(100 − 64) = 6 cm.
  • Medium: Chord 24 cm, r = 5 cm: R = √(144 + 25) = 13 cm.
  • Exam-type: Always R > r, since the chord lies in the bigger circle.
Remember

The radius to the point of contact bisects the chord because it is perpendicular to it.

Common mistake

Giving the half chord as the answer when the full chord is asked; multiply by 2.

Memory clue

"Big radius square minus small radius square, root, then double."

Practice questions

Easy
  1. 1Two concentric circles with centre O have radii 17 cm and 3 cm. A chord VW of the larger circle touches the smaller circle at L. Find OL (in cm).
  2. 2Two concentric circles with centre O have radii 40 cm and 5 cm. A chord AB of the larger circle touches the smaller circle at Y. Find OY (in cm).
  3. 3Two concentric circles have centre O. A chord YB of the larger circle touches the smaller circle at F. If YF = 19 cm, find YB (in cm).
Medium
  1. 4A chord of a circle of radius 58 cm is 84 cm long and it touches a concentric smaller circle. Find the radius of the smaller circle (in cm).
  2. 5A chord of a circle of radius 25 cm is 40 cm long and it touches a concentric smaller circle. Find the radius of the smaller circle (in cm).
  3. 6Two concentric circles have radii 17 cm and 8 cm. Find the length of the chord of the larger circle which touches the smaller circle (in cm).
Challenging
  1. 7A chord of the larger of two concentric circles is 68 cm long and touches the smaller circle. Find the area of the ring between the two circles (in cm², in terms of π).
  2. 8Two concentric circles with centre O have radii 10 cm and 6 cm. A chord ZC of the larger circle touches the smaller circle. Find the perimeter of the triangle OZC (in cm).
Show answers (try them yourself first)
  1. 13
  2. 25
  3. 338
  4. 440
  5. 515
  6. 630
  7. 71156π
  8. 836

Parallel tangents and tangent configurations

From daily life

A ball rolling between two straight walls touches both walls; the walls are exactly one ball-width apart.

Simple definition

The tangents at the two ends of a diameter are parallel, because both are perpendicular to the diameter. The distance between them is the diameter 2r. A circle touching two parallel lines has its centre midway between them.

OPQPQ = 2r= distancebetween tangents

In the figure: Tangents at P and Q, the ends of a diameter, are parallel and 2r apart.

Rule 1: Parallel tangents — Use the diameter as the distance between them.

  • Easy: r = 4 cm: the parallel tangents are 8 cm apart.
  • Medium: Two parallel lines 12 cm apart: the touching circle has r = 6 cm.
  • Exam-type: Its centre lies on the line 6 cm from each.

Rule 2: Circle in a square — A circle touching all four sides of a square has diameter equal to the side.

  • Easy: Square of side 10 cm: the circle has r = 5 cm.
  • Medium: Check: the distance between opposite sides is 10 = 2r.
  • Exam-type: A tangent is perpendicular to the diameter drawn to its point of contact.
Remember

Tangent at one end of a diameter plus tangent at the other end: always parallel.

Common mistake

Taking the radius, not the diameter, as the distance between the parallel tangents.

Memory clue

"Parallel tangents are a whole diameter apart."

Practice questions

Easy
  1. 1Two parallel tangents to a circle are 102 cm apart. Find the radius of the circle (in cm).
  2. 2The diameter of a circle is 48 cm. Tangents are drawn at the two ends of this diameter. Find the distance between the two tangents (in cm).
  3. 3Two parallel tangents to a circle are 26 cm apart. Find the radius of the circle (in cm).
Medium
  1. 4A tangent YS touches a circle with centre O at M. It meets two parallel tangents of the circle at Y and S, where ∠YOS = 90°. If YM = 36 cm and SM = 144 cm, find the radius of the circle (in cm).
  2. 5Two parallel tangents to a circle with centre O are touched by a third tangent RT (with point of contact Y) at R and T. If ∠ORT = 80°, find ∠OTR (in degrees).
  3. 6Two parallel tangents to a circle with centre O are touched by a third tangent RV (with point of contact W) at R and V. If ∠ORV = 80°, find ∠OVR (in degrees).
Challenging
  1. 7A tangent TX touches a circle with centre O at Y and meets two parallel tangents at T and X with ∠TOX = 90°. If TY = 36 cm and XY = 81 cm, find the area of the triangle TOX (in cm²).
  2. 8Tangents at S and G to a circle with centre O are parallel, and a third tangent meets them at S and G with ∠SOG = 90°. If OS = 3 cm and OG = 4 cm, find the area of the triangle SOG (in cm²).
Show answers (try them yourself first)
  1. 151
  2. 248
  3. 313
  4. 472
  5. 510
  6. 610
  7. 73159
  8. 86