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CBSE · Class 10 · Mathematics

Circles 10.3: Number of Tangents from a Point on a Circle

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

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Number of tangents from a point

From daily life

Standing inside a round room you cannot draw a line that only grazes the wall, but from outside you can look along two grazing lines.

Simple definition

From a point inside the circle no tangent can be drawn. From a point on the circle exactly one tangent can be drawn. From a point outside (an external point) exactly two tangents can be drawn.

Oinside: 0on circle: 1outside: 2

In the figure: Inside: 0 tangents. On the circle: 1. Outside: 2.

Rule 1: Compare OP with r — The distance of P from the centre decides the number.

  • Easy: r = 5 cm, OP = 3 cm: P is inside, so 0 tangents.
  • Medium: r = 5 cm, OP = 5 cm: P is on the circle, so 1 tangent.
  • Exam-type: r = 5 cm, OP = 9 cm: P is outside, so 2 tangents.

Rule 2: Why not from inside? — Any line through an inside point cuts the circle twice.

  • Easy: So every line through an inside point is a secant.
  • Medium: From OP = 13 cm, r = 5 cm: two tangents, each of length 12 cm.
  • Exam-type: The two tangents lie on the two sides of the line OP.
Remember

Only the outside point gives two tangents, and they are mirror images about OP.

Common mistake

Saying a point on the circle has two tangents; it has exactly one.

Memory clue

"Inside none, on it one, outside two."

Practice questions

Easy
  1. 1A point C is 22 cm from the centre of a circle of radius 22 cm. How many tangents can be drawn from C to the circle?
  2. 2A point C is 8 cm from the centre of a circle of radius 8 cm. How many tangents can be drawn from C to the circle?
  3. 3A point W is 36 cm from the centre of a circle of radius 36 cm. How many tangents can be drawn from W to the circle?
Medium
  1. 4Three points are at distances 21 cm, 25 cm and 24 cm from the centre of a circle of radius 19 cm. What is the total number of tangents that can be drawn to the circle from these three points (counting each point separately)?
  2. 5The diameter of a circle is 54 cm. How many tangents can be drawn from a point M at a distance of 4 cm from the centre?
  3. 6Three points are at distances 6 cm, 2 cm and 6 cm from the centre of a circle of radius 6 cm. What is the total number of tangents that can be drawn to the circle from these three points (counting each point separately)?
Challenging
  1. 7Four points are at distances 12 cm, 4 cm, 6 cm and 5 cm from the centre of a circle of radius 12 cm. From how many of these points can exactly two tangents be drawn?
  2. 8Four points are at distances 3 cm, 19 cm, 36 cm and 38 cm from the centre of a circle of radius 25 cm. From how many of these points is it impossible to draw a tangent?
Show answers (try them yourself first)
  1. 11
  2. 21
  3. 31
  4. 46
  5. 50
  6. 62
  7. 70
  8. 82

Equal lengths of tangents

From daily life

The two ropes pulled tight from a tent peg to a round pillar, each just touching the pillar, are equally long.

Simple definition

The two tangents from an external point are equal in length: PA = PB. Right triangles OAP and OBP have OA = OB (radii), a common hypotenuse OP and a right angle each, so they are congruent.

Formula: PA = PB = √(OP² − r²)

OPAB12125PA = PBOP = 13, r = 5PA = √(169 − 25)= 12

In the figure: OP = 13, r = 5: both tangents are √(169 − 25) = 12 long.

Live figure

Equal lengths of tangents

In the app you can move this figure with your finger.

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Rule 1: Use PA = PB — Whatever you find for one tangent holds for the other.

  • Easy: PA = 7 cm gives PB = 7 cm.
  • Medium: OP = 17 cm, r = 8 cm: PA = PB = 15 cm.
  • Exam-type: Triangle PAB is isosceles with PA = PB.

Rule 2: Why the tangents are equal — Match the two right triangles by RHS: right angle, hypotenuse, one side.

  • Easy: OA = OB (radii) and OP common.
  • Medium: So triangle OAP is congruent to triangle OBP, hence PA = PB.
  • Exam-type: The line OP also bisects angle APB and angle AOB.
Remember

Equal tangent lengths work only for tangents drawn from the same external point.

Common mistake

Assuming tangents from two different points are equal.

Memory clue

"From one point, two tangents, one length."

Practice questions

Easy
  1. 1Two tangents MQ and MX are drawn from a point M to a circle and MQ + MX = 52 cm. Find MQ (in cm).
  2. 2AV and AF are the two tangents from an external point A to a circle. If AV = 29 cm, find AF (in cm).
  3. 3GU and GD are the two tangents from an external point G to a circle. If GD = 33 cm, find GU (in cm).
Medium
  1. 4LV and LE are the two tangents from an external point L to a circle. If LV = 56 cm, find LE (in cm).
  2. 5From an external point X two tangents XQ and XU are drawn to a circle with XQ = (2x + 9) cm and XU = (5x − 9) cm; find the length of each tangent (in cm).
  3. 6From an external point U two tangents UQ and UX are drawn to a circle with UQ = (x + 3) cm and UX = (6x − 12) cm; find x.
Challenging
  1. 7From an external point E two tangents EK and EB are drawn to a circle with EK = (3x + 5) cm and EB = (2x + 18) cm; find the length of each tangent (in cm).
  2. 8Tangents DM and DE are drawn from D to a circle with ∠MDE = 90° and DM = 11 cm. Find the length of the chord ME (in cm).
Show answers (try them yourself first)
  1. 126
  2. 229
  3. 333
  4. 456
  5. 521
  6. 63
  7. 744
  8. 811√2

Angle between two tangents

From daily life

The two tangents and two radii make a kite whose two side angles are right angles, so the other two angles must make up a straight half-turn.

Simple definition

In the quadrilateral OAPB the angles at A and B are 90° each, so ∠APB + ∠AOB = 180°. The line OP bisects both angles, so ∠APO = ½ ∠APB and ∠AOP = 90° − ∠APO.

Formula: ∠APB + ∠AOB = 180°

OPAB80°100°angle P + angle O= 180°OP bisects both

In the figure: ∠APB = 80° gives ∠AOB = 100°.

Rule 1: Use the 180° relation — Subtract the known angle from 180°.

  • Easy: ∠AOB = 120°: ∠APB = 180° − 120° = 60°.
  • Medium: ∠APB = 80°: ∠AOB = 100°.
  • Exam-type: The two angles are supplementary.

Rule 2: Use the bisector OP — Half of the angle at P goes into the right triangle OAP.

  • Easy: ∠APB = 70°: ∠APO = 35°, so ∠AOP = 55°.
  • Medium: Check: ∠AOB = 110° and 70° + 110° = 180°.
  • Exam-type: If ∠APB = 60°, triangle PAB is equilateral since PA = PB; PA = 8 cm gives AB = 8 cm.
Remember

Angle at the centre and angle at the outside point always add up to 180°.

Common mistake

Subtracting from 90° instead of 180° for the angle at the centre.

Memory clue

"Outside angle plus centre angle makes 180."

Practice questions

Easy
  1. 1Tangents MU and MD are drawn from M to a circle with centre O and ∠UMD = 150°. Find ∠OMU (in degrees).
  2. 2Tangents HR and HU are drawn from H to a circle with centre O. If ∠RHU = 65°, find ∠ROU (in degrees).
  3. 3Tangents BZ and BC are drawn from B to a circle with centre O. If ∠ZBC = 160°, find ∠ZOC (in degrees).
Medium
  1. 4Tangents ST and SA are drawn from S to a circle with centre O and ∠TOA = 130°. Find ∠STA (in degrees).
  2. 5Tangents PG and PK are drawn from P to a circle with ∠GPK = 90°. Find ∠PGK (in degrees).
  3. 6Tangents TA and TX are drawn from T to a circle with centre O and ∠AOX = 140°. Find ∠TAX (in degrees).
Challenging
  1. 7Tangents QG and QR are drawn from Q to a circle with centre O such that ∠GQR : ∠GOR = 1 : 2. Find ∠GQR (in degrees).
  2. 8Tangents VS and VE are drawn from V to a circle with centre O. If ∠SOE = (4x + 19)° and ∠SVE = (x + 56)°, find ∠SVE (in degrees).
Show answers (try them yourself first)
  1. 175
  2. 2115
  3. 320
  4. 465
  5. 545
  6. 670
  7. 760
  8. 877

Quadrilaterals and triangles around a circle

From daily life

A round table fitted inside a four-sided frame touches each side once. The side lengths of the frame then follow a neat rule.

Simple definition

If a quadrilateral ABCD has all four sides touching a circle, the sums of opposite sides are equal: AB + CD = AD + BC. This follows because the two tangents from each vertex are equal. In a triangle with an incircle, the tangent lengths from each vertex are equal too.

Formula: AB + CD = AD + BC

ABC755337AB + CD = AD + BC(quadrilateral)AB = 10, BC = 12,CA = 8: tangents3, 7, 5

In the figure: Triangle with sides 10, 12, 8: tangent lengths from A, B, C are 3, 7, 5.

Rule 1: Opposite sides of a quadrilateral — Add each pair of opposite sides.

  • Easy: AB = 7, CD = 9, AD = 8: BC = 7 + 9 − 8 = 8.
  • Medium: Check: 7 + 9 = 16 and 8 + 8 = 16.
  • Exam-type: A parallelogram around a circle is a rhombus: AB = CD, AD = BC, so AB = BC.

Rule 2: Triangle with an incircle — Give each vertex its own tangent length and match the sides.

  • Easy: AB = 10, BC = 12, CA = 8: AF = AE = 3, BF = BD = 7, CD = CE = 5.
  • Medium: Check: 3 + 7 = 10, 7 + 5 = 12, 5 + 3 = 8.
  • Exam-type: Each side is the sum of two tangent lengths.
Remember

Name the tangent length at each vertex with one letter, then write each side as a sum.

Common mistake

Pairing adjacent sides instead of opposite sides; AB + CD matches AD + BC.

Memory clue

"Opposite sides add to the same total."

Practice questions

Easy
  1. 1A quadrilateral ABCD circumscribes a circle. If AB = 8 cm, BC = 20 cm, DA = 3 cm, find CD (in cm).
  2. 2A quadrilateral ABCD circumscribes a circle. If AB = 7 cm, BC = 22 cm, DA = 15 cm, find CD (in cm).
  3. 3A quadrilateral ABCD circumscribes a circle with AB + CD = 35 cm. Find the perimeter of ABCD (in cm).
Medium
  1. 4A triangle ABC circumscribes a circle that touches AB, BC and CA at F, D and E. If AF = 17 cm, BD = 18 cm and CE = 17 cm, find CA (in cm).
  2. 5A parallelogram PQRS circumscribes a circle and PQ = 38 cm; find QR (in cm).
  3. 6A triangle ABC circumscribes a circle that touches AB, BC and CA at F, D and E. If AF = 6 cm, BD = 14 cm and CE = 7 cm, find the perimeter of the triangle ABC (in cm).
Challenging
  1. 7A circle is inscribed in a triangle ABC with AB = 12 cm, BC = 18 cm and CA = 20 cm. Find the length of the tangent from C to the circle (in cm).
  2. 8A quadrilateral ABCD circumscribes a circle with AB = (x + 5) cm, BC = (x + 10) cm, CD = (2x + 15) cm and DA = (x + 13) cm. Find the perimeter of ABCD (in cm).
Show answers (try them yourself first)
  1. 115
  2. 230
  3. 370
  4. 434
  5. 538
  6. 654
  7. 713
  8. 858

Applications of equal tangents

From daily life

A belt passing round two pulleys and a shape hugging a circle are all solved by replacing one tangent by its equal partner.

Simple definition

Mark equal tangent lengths on the figure, then use the right triangle OAP and the angle relations. If a third tangent XY touches the circle at C, then XA = XC and YB = YC, so the perimeter of triangle PXY is PA + PB = 2PA.

PABCXYPA = 12perimeter = PA + PB= 24

In the figure: Tangent XY at C: the perimeter of triangle PXY is PA + PB = 24 when PA = 12.

Rule 1: Third tangent — Replace XC by XA and YC by YB.

  • Easy: XA = XC and YB = YC.
  • Medium: Perimeter of PXY = PX + XC + CY + YP = PA + PB.
  • Exam-type: PA = 12 cm gives perimeter 24 cm.

Rule 2: Kites and squares — Use right angles at A and B and the equal radii.

  • Easy: ∠APB = 90°, r = 6 cm: OAPB is a square, so PA = 6 cm.
  • Medium: Then OP = 6√2 ≈ 8.49 cm, the diagonal of the square.
  • Exam-type: OP = 13, r = 5: area of OAPB = 2 × ½ × 5 × 12 = 60 square units.
Remember

Check every answer by finding some length two ways, for example PA from OP and r.

Common mistake

Forgetting that XA = XC; the third tangent’s two parts match the tangents from X and Y.

Memory clue

"Swap the tangent for its twin, then add."

Practice questions

Easy
  1. 1Tangents AQ and AH are drawn from A to a circle of radius 24 cm with centre O. If OA = 51 cm, find AQ (in cm).
  2. 2Tangents PW and PU are drawn from P to a circle of radius 16 cm with centre O, and PW = 30 cm. Find the area of the quadrilateral OWPU (in cm²).
  3. 3Tangents FV and FU are drawn from F to a circle of radius 48 cm with centre O, and FV = 20 cm. Find the area of the quadrilateral OVFU (in cm²).
Medium
  1. 4A circle touches all three sides of a triangle ABC right-angled at B, with AB = 40 cm and BC = 42 cm. Find the radius of the circle (in cm).
  2. 5Tangents EH and ER are drawn from E to a circle with EH = 38 cm. A third tangent touches the circle at F on the minor arc HR and meets EH at S and ER at M. Find the perimeter of the triangle ESM (in cm).
  3. 6Tangents TL and TE are drawn from T to a circle with TL = 9 cm. A third tangent touches the circle at V on the minor arc LE and meets TL at U and TE at W. Find the perimeter of the triangle TUW (in cm).
Challenging
  1. 7Tangents YT and YW to a circle of radius 14 cm with centre O are inclined at an angle of 60°. Find YT (in cm).
  2. 8Tangents WD and WG are drawn from W to a circle of radius 15 cm with centre O, and WD = 20 cm. Find the length of the chord DG (in cm).
Show answers (try them yourself first)
  1. 145
  2. 2480
  3. 3960
  4. 412
  5. 576
  6. 618
  7. 714√3
  8. 824