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Circles · Class 10 Mathematics: 114 MCQs with answers

All 114 multiple-choice questions from this kit, free and with no sign-up.

Full kit · everything Flashcards · 44 Quick Revision · 35

Warm-up · 21 questions

Q1. A line meeting a circle at two points is called a

Beginner
A tangent B radius C secant D diameter
Show answer & explanation

secant

It cuts through.

Q2. A line meeting a circle at exactly one point is called a

Beginner
A tangent B secant C chord D diameter
Show answer & explanation

tangent

It touches.

Q3. The single point a tangent shares with its circle is the

Beginner
A point of contact B centre C midpoint D vertex
Show answer & explanation

point of contact

That is its name.

Q4. The greatest number of points a straight line can share with a circle is

Beginner
A two B one C three D unlimited
Show answer & explanation

two

A secant is the extreme case.

Q5. The part of a secant lying inside the circle is a

Beginner
A tangent B radius C arc D chord
Show answer & explanation

chord

It joins the two crossing points.

Q6. The tangent at a point of a circle is perpendicular to

Beginner
A every radius B the diameter nearest it C the nearest chord D the radius through the point of contact
Show answer & explanation

the radius through the point of contact

Theorem 1.

Q7. The right angle in a tangent figure is marked at the

Beginner
A point of contact B centre C external point D midpoint of the radius
Show answer & explanation

point of contact

Where the tangent touches.

Q8. A point 10 cm from the centre of a circle of radius 6 cm has a tangent of length

Beginner
A 4 cm B 16 cm C 8 cm D $\sqrt{136}$ cm
Show answer & explanation

8 cm

$\sqrt{100-36} = 8$.

Q9. How many tangents can be drawn through a point INSIDE a circle?

Beginner
A none B one C two D infinitely many
Show answer & explanation

none

Every line through it must leave the circle on both sides.

Q10. The single tangent at a point of a circle is the line through that point perpendicular to

Beginner
A the nearest diameter B the radius drawn to it C any chord D the tangent at the opposite end
Show answer & explanation

the radius drawn to it

The line perpendicular to the radius there.

Q11. Two tangents can be drawn to a circle from a point whose distance from the centre is

Beginner
A equal to the radius B less than the radius C zero D greater than the radius
Show answer & explanation

greater than the radius

One on each side.

Q12. A tangent touches a circle at $P$ and the centre is $O$. The angle $\angle OPT$, with $T$ on the tangent, is

Beginner
A 45° B 60° C 90° D unknown without more data
Show answer & explanation

90°

Theorem 1 gives it directly.

Q13. The word 'tangent' comes from a Latin verb meaning

Beginner
A to cut B to touch C to turn D to measure
Show answer & explanation

to touch

Tangere — the same root as 'tangible'.

Q14. A wheel resting on flat ground touches it along

Beginner
A a secant B a chord C a tangent D a diameter
Show answer & explanation

a tangent

One point of contact.

Q15. Distinguish between a secant and a tangent, and name the special point in each case.

Beginner
Show answer & explanation

—

Q16. State Theorem 1 of this chapter precisely.

Beginner
Show answer & explanation

—

Q17. Drawing two tangents to a circle from a single outside point, their lengths turn out to be

Beginner
A in the ratio of the radii B equal C always different D equal to the radius
Show answer & explanation

equal

Theorem 2.

Q18. A circle of radius 20 cm has an external point 29 cm away. Each tangent length is

Beginner
A 9 cm B 35 cm C $\sqrt{1241}$ cm D 21 cm
Show answer & explanation

21 cm

$\sqrt{841-400} = 21$.

Q19. If $PA$ and $PB$ are tangents from $P$ and $PA = 9$ cm, then $PB$ is

Beginner
A 18 cm B 9 cm C 4.5 cm D not determined
Show answer & explanation

9 cm

Theorem 2 makes them equal.

Q20. A circle touches all three sides of a triangle. That circle is called the

Beginner
A circumcircle B excircle C incircle D nine-point circle
Show answer & explanation

incircle

It sits inside, touching each side.

Q21. State Theorem 2 of this chapter and name the congruence criterion used to prove it.

Beginner
Show answer & explanation

—

Building up · 40 questions

Q22. In the proof of Theorem 1, every point of the tangent other than the point of contact lies

Intermediate
A inside the circle B on the circle C at the centre D outside the circle
Show answer & explanation

outside the circle

Otherwise the line would meet the circle more than once.

Q23. The proof concludes that $OP$ is perpendicular because

Intermediate
A the angle is given B the shortest distance from a point to a line is the perpendicular one C all radii are equal D the triangle is isosceles
Show answer & explanation

the shortest distance from a point to a line is the perpendicular one

$OP$ was shown to be the shortest such distance.

Q24. In triangle $OAP$, where $PA$ is a tangent touching at $A$, the hypotenuse is

Intermediate
A $OA$ B $OP$ C $AP$ D any of them
Show answer & explanation

$OP$

It faces the right angle at $A$.

Q25. The length of a tangent from a point at distance $d$ to a circle of radius $r$ is

Intermediate
A $\sqrt{d^2 - r^2}$ B $\sqrt{d^2 + r^2}$ C $d - r$ D $\sqrt{r^2 - d^2}$
Show answer & explanation

$\sqrt{d^2 - r^2}$

Pythagoras in the right triangle.

Q26. A point is 13 cm from the centre of a circle of radius 5 cm. The tangent length is

Intermediate
A 8 cm B 12 cm C 18 cm D $\sqrt{194}$ cm
Show answer & explanation

12 cm

$\sqrt{169-25} = 12$.

Q27. A tangent 24 cm long is drawn to a circle of radius 7 cm. The point is

Intermediate
A 31 cm from the centre B 17 cm from the centre C $\sqrt{527}$ cm from the centre D 25 cm from the centre
Show answer & explanation

25 cm from the centre

$\sqrt{49+576} = 25$.

Q28. A circle of radius 8 cm has a tangent of length 15 cm from a point. That point is

Intermediate
A 23 cm from the centre B 7 cm from the centre C 17 cm from the centre D $\sqrt{161}$ cm from the centre
Show answer & explanation

17 cm from the centre

$\sqrt{64+225} = 17$.

Q29. The tangents at the two ends of a diameter are

Intermediate
A perpendicular B equal in length C parallel D coincident
Show answer & explanation

parallel

Both are perpendicular to the same straight line.

Q30. Which fact makes those two tangents parallel?

Intermediate
A two lines perpendicular to the same line are parallel B all tangents are parallel C the diameter bisects them D they are equal
Show answer & explanation

two lines perpendicular to the same line are parallel

It is the standard parallel-lines result.

Q31. Assertion (A): The radius drawn to the point of contact is the shortest segment from the centre to the tangent line. Reason (R): The shortest distance from a point to a line is measured along the perpendicular.

Intermediate
A Both A and R are true, and R is the correct explanation of A B Both A and R are true, but R is not the correct explanation of A C A is true but R is false D A is false but R is true
Show answer & explanation

Both A and R are true, and R is the correct explanation of A

Theorem 1 says that radius IS the perpendicular, so R explains A exactly.

Q32. The converse of Theorem 1 states that a line through a point of a circle is a tangent if it is

Intermediate
A parallel to a diameter B shorter than the radius C perpendicular to the radius at that point D equal to the radius
Show answer & explanation

perpendicular to the radius at that point

That is how you prove a line is a tangent.

Q33. The 'length of the tangent' from a point means the distance from that point to

Intermediate
A the centre B the far side of the circle C the nearest chord D the point of contact
Show answer & explanation

the point of contact

The line itself is infinite; only this segment has a length.

Q34. Marking the right angle at the CENTRE instead of the point of contact would

Intermediate
A make no difference B be an equivalent statement C only affect the diagram D make the whole solution wrong
Show answer & explanation

make the whole solution wrong

It changes which side is the hypotenuse.

Q35. A line drawn through a point inside a circle must be

Intermediate
A a tangent B a secant C outside the circle D a radius
Show answer & explanation

a secant

It has to leave the circle on both sides.

Q36. In a tangent figure, which theorem allows Pythagoras to be used at all?

Intermediate
A all radii are equal B the angle sum of a triangle C the tangent is perpendicular to the radius D the converse of Pythagoras
Show answer & explanation

the tangent is perpendicular to the radius

Pythagoras needs a right angle, and Theorem 1 supplies it.

Q37. If a line is at a distance from the centre EQUAL to the radius, the line is

Intermediate
A a secant B a line that misses the circle C a diameter D a tangent
Show answer & explanation

a tangent

Exactly one point of it is at distance $r$.

Q38. A circle has radius 6 cm and a straight line runs 7 cm from its centre. Their number of common points is

Intermediate
A one B two C zero D three
Show answer & explanation

zero

Every point of it is further than $r$.

Q39. If a line is at a distance from the centre LESS than the radius, it is

Intermediate
A a tangent B a secant C a line that misses D impossible
Show answer & explanation

a secant

It cuts the circle twice.

Q40. A point is 17 cm from the centre of a circle of radius 8 cm. Find the length of the tangent from that point.

Intermediate
Show answer & explanation

—

Q41. A tangent of length 12 cm is drawn to a circle of radius 5 cm. How far is the external point from the centre?

Intermediate
Show answer & explanation

—

Q42. State how many tangents can be drawn to a circle through a point in each of the three possible positions, with a reason for each.

Intermediate
Show answer & explanation

—

Q43. Explain why the line from the centre to an external point, and not the radius, is the hypotenuse of the tangent triangle.

Intermediate
Show answer & explanation

—

Q44. State the converse of Theorem 1 and say when it is used.

Intermediate
Show answer & explanation

—

Q45. The congruence criterion used to prove Theorem 2 is

Intermediate
A SSS B ASA C RHS D AAA
Show answer & explanation

RHS

Two right triangles with equal hypotenuses and one equal leg.

Q46. In that proof, the equal legs of the two right triangles are

Intermediate
A the two tangents B the two radii C the two hypotenuses D the chord of contact
Show answer & explanation

the two radii

$OA = OB$, radii of the same circle.

Q47. In that proof, the common hypotenuse is

Intermediate
A $OA$ B $PA$ C $OP$ D $AB$
Show answer & explanation

$OP$

It belongs to both triangles.

Q48. The right angles used in the proof come from

Intermediate
A the angle sum property B Theorem 1 C the given data D Pythagoras
Show answer & explanation

Theorem 1

The tangent is perpendicular to the radius at the point of contact.

Q49. The step that turns the congruence into $PA = PB$ is

Intermediate
A RHS B Pythagoras C CPCT D the converse of Theorem 1
Show answer & explanation

CPCT

Corresponding parts of congruent triangles are equal.

Q50. The segment $OP$ bisects

Intermediate
A only the angle between the tangents B only the angle between the radii C both the angle between the tangents and the angle between the radii D neither of those angles
Show answer & explanation

both the angle between the tangents and the angle between the radii

Both follow by CPCT from the same congruence.

Q51. A circle of radius 7 cm has an external point 25 cm from its centre. Each tangent length is

Intermediate
A 18 cm B 32 cm C $\sqrt{674}$ cm D 24 cm
Show answer & explanation

24 cm

$\sqrt{625-49} = 24$.

Q52. In a quadrilateral $ABCD$ circumscribing a circle, which equation holds?

Intermediate
A $AB + BC = CD + DA$ B $AB + CD = BC + DA$ C $AB = CD$ D $AB \times CD = BC \times DA$
Show answer & explanation

$AB + CD = BC + DA$

Opposite sides have equal sums.

Q53. A triangle has sides 14, 10 and 12 cm. Its semiperimeter is

Intermediate
A 36 cm B 12 cm C 9 cm D 18 cm
Show answer & explanation

18 cm

Half of $14+10+12 = 36$.

Q54. Assertion (A): The tangents drawn from an external point to a circle are equal in length. Reason (R): The two right triangles formed are congruent by RHS, having equal radii and a common hypotenuse.

Intermediate
A Both A and R are true, and R is the correct explanation of A B Both A and R are true, but R is not the correct explanation of A C A is true but R is false D A is false but R is true
Show answer & explanation

Both A and R are true, and R is the correct explanation of A

R sets out exactly the congruence from which A follows by CPCT.

Q55. The first practical step in a tangent-figure question is to

Intermediate
A measure the diagram B apply Pythagoras C give equal tangent lengths from each vertex the same letter D find the area
Show answer & explanation

give equal tangent lengths from each vertex the same letter

The labelling is what turns the figure into equations.

Q56. The two tangents from an external point, together with the chord of contact, form a triangle that is always

Intermediate
A isosceles B equilateral C right-angled D scalene
Show answer & explanation

isosceles

The two tangents are equal.

Q57. A circle of radius 9 cm has tangents of length 12 cm from a point. That point is

Intermediate
A 15 cm from the centre B 21 cm from the centre C 3 cm from the centre D $\sqrt{63}$ cm from the centre
Show answer & explanation

15 cm from the centre

$\sqrt{81+144} = 15$.

Q58. In the kite $OAPB$, the two right angles are at

Intermediate
A $O$ and $P$ B $A$ and $O$ C $P$ only D $A$ and $B$
Show answer & explanation

$A$ and $B$

They are the points of contact.

Q59. Name three results, besides $PA = PB$, that follow from the same congruence.

Intermediate
Show answer & explanation

—

Q60. A circle has radius 7 cm and a point $P$ lies 25 cm from its centre. Find the length of each tangent from $P$.

Intermediate
Show answer & explanation

—

Q61. A quadrilateral circumscribes a circle. Three consecutive sides are 6 cm, 7 cm and 4 cm. Find the fourth side.

Intermediate
Show answer & explanation

—

Topper level · 53 questions

Q62. Assertion (A): A tangent can be drawn to a circle from any point of the plane. Reason (R): A point outside a circle has exactly two tangents through it.

Advanced
A Both A and R are true, and R is the correct explanation of A B Both A and R are true, but R is not the correct explanation of A C A is true but R is false D A is false but R is true
Show answer & explanation

Both A and R are true, but R is not the correct explanation of A

R is true, but A is false — a point inside the circle has no tangent at all.

Q63. To PROVE that a given line touching a circle at $P$ is a tangent, you would use

Advanced
A the converse of Theorem 1 B Theorem 1 itself C Pythagoras alone D the angle sum property
Show answer & explanation

the converse of Theorem 1

The theorem uses tangency; the converse establishes it.

Q64. A tangent is best described as the limiting position of

Advanced
A a radius B a chord through the centre C a diameter D a secant whose two points of intersection coincide
Show answer & explanation

a secant whose two points of intersection coincide

Slide the secant outwards until the two points merge.

Q65. Which point of a tangent line is nearest to the centre of the circle?

Advanced
A a point far to the left B the point of contact C the midpoint of the visible part D they are all equally near
Show answer & explanation

the point of contact

The perpendicular distance is the shortest, and it lands at the contact point.

Q66. A circle has radius 9 cm. A tangent is drawn from a point 41 cm from the centre. Its length is

Advanced
A 32 cm B 50 cm C $\sqrt{1762}$ cm D 40 cm
Show answer & explanation

40 cm

$\sqrt{1681 - 81} = 40$.

Q67. If the radius is 5 cm and a point is 5 cm from the centre, the tangent length from it is

Advanced
A 5 cm B 0, since the point lies on the circle C 10 cm D undefined
Show answer & explanation

0, since the point lies on the circle

The point of contact IS the point, so the distance between them is zero.

Q68. Can a tangent be drawn from a point 3 cm from the centre of a circle of radius 5 cm?

Advanced
A no, the point is inside the circle B yes, one C yes, two D yes, infinitely many
Show answer & explanation

no, the point is inside the circle

$3 < 5$, so the point is inside.

Q69. Two circles of the same centre are drawn. A tangent to the smaller one is, for the larger one,

Advanced
A a secant B also a tangent C a line that misses it D a diameter
Show answer & explanation

a secant

Its distance from the centre equals the smaller radius, which is less than the larger.

Q70. A ball whirled on a string flies off, when released, along

Advanced
A the radius B the tangent at that point C a chord D a spiral
Show answer & explanation

the tangent at that point

Its velocity at that instant points along the tangent.

Q71. Assertion (A): A line meeting a circle at exactly one point is perpendicular to the radius at that point. Reason (R): Every other point of such a line lies inside the circle.

Advanced
A Both A and R are true, and R is the correct explanation of A B Both A and R are true, but R is not the correct explanation of A C A is true but R is false D A is false but R is true
Show answer & explanation

A is true but R is false

The other points lie outside the circle, which is what makes the contact point the nearest one. The assertion holds, the reason is false.

Q72. Assertion (A): The radius is the hypotenuse of the right triangle formed by a tangent, the radius and the line to an external point. Reason (R): The right angle in that triangle is at the point of contact.

Advanced
A Both A and R are true, and R is the correct explanation of A B Both A and R are true, but R is not the correct explanation of A C A is true but R is false D A is false but R is true
Show answer & explanation

A is false but R is true

R is true, and it is exactly why A is false — the hypotenuse faces the right angle, so it is the line from the centre to the external point.

Q73. Prove that the tangent meets the radius at its point of contact squarely, at 90°.

Advanced
Show answer & explanation

—

Q74. Prove that the two tangents drawn where a diameter meets the circle run parallel.

Advanced
Show answer & explanation

—

Q75. A line lies at a distance $p$ from the centre of a circle of radius $r$. Describe the three cases.

Advanced
Show answer & explanation

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Q76. (a) Explain, with reasons, the three ways in which a line and a circle can be related. (b) Prove that the tangent meets the radius at its point of contact squarely, at 90°. (c) Explain which step of your proof would fail if the line were a secant instead.

Advanced
Show answer & explanation

—

Q77. (a) Explain how Theorem 1 makes Pythagoras' theorem available in circle problems, naming the right angle and the hypotenuse. (b) A point $P$ is 25 cm from the centre $O$ of a circle of radius 7 cm, and $PA$ is a tangent touching at $A$. Find $PA$. (c) A second circle, concentric with the first, has radius 24 cm. Show that $PA$ is itself the radius of a circle centred at $P$ that touches the first circle, and state the connection with (b).

Advanced
Show answer & explanation

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Q78. (a) State and prove that the tangents at the ends of a diameter are parallel. (b) State the converse of Theorem 1. (c) A line $l$ meets a circle of centre $O$ at a single point $T$, and it is given that $\angle OTl = 90°$. Explain which result is being used and what may be concluded.

Advanced
Show answer & explanation

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Q79. (a) Describe how many tangents pass through a point in each position relative to a circle, justifying each count. (b) A circle has radius 6 cm. Decide, with reasons, how many tangents pass through a point 4 cm from the centre, and how many through a point 10 cm from the centre, giving the tangent length where one exists. (c) Explain what happens to the tangent length as the external point moves towards the circle.

Advanced
Show answer & explanation

—

Q80. The quadrilateral $OAPB$ formed by the two radii and the two tangents is a

Advanced
A square B rectangle C kite D trapezium
Show answer & explanation

kite

$OA = OB$ and $PA = PB$, in adjacent pairs.

Q81. $OP$ is, with respect to the chord of contact $AB$,

Advanced
A its perpendicular bisector B parallel to it C equal to it D unrelated
Show answer & explanation

its perpendicular bisector

It is the axis of symmetry of the kite $OAPB$.

Q82. That quadrilateral result is proved by applying Theorem 2

Advanced
A once at each of the four vertices B once in total C twice D eight times
Show answer & explanation

once at each of the four vertices

Each vertex is an external point.

Q83. A quadrilateral circumscribes a circle and three of its sides are 6, 7 and 4 cm in that order. The fourth side is

Advanced
A 5 cm B 3 cm C 9 cm D 17 cm
Show answer & explanation

3 cm

$6 + 4 = 7 + x$ gives $x = 3$.

Q84. A parallelogram circumscribing a circle must be a

Advanced
A rectangle B rhombus C square only D trapezium
Show answer & explanation

rhombus

Equal opposite sides plus equal opposite sums force all four to be equal.

Q85. Can a rectangle measuring 3 cm by 5 cm circumscribe a circle?

Advanced
A yes, always B yes, if the circle is small enough C no, because $3+3 \ne 5+5$ D only if it is rotated
Show answer & explanation

no, because $3+3 \ne 5+5$

The opposite-sides condition fails.

Q86. For a triangle with sides $a$, $b$, $c$ and semiperimeter $s$, the incircle tangent length from the vertex opposite $a$ is

Advanced
A $s - a$ B $s + a$ C $\frac{a}{2}$ D $s$
Show answer & explanation

$s - a$

Subtract the opposite side from the semiperimeter.

Q87. For that triangle, the tangent length from the vertex opposite the 14 cm side is

Advanced
A 8 cm B 6 cm C 14 cm D 4 cm
Show answer & explanation

4 cm

$s - a = 18 - 14 = 4$.

Q88. Assertion (A): Theorem 2 can be proved using the SSS criterion. Reason (R): The two triangles share the segment $OP$ and have equal radii.

Advanced
A Both A and R are true, and R is the correct explanation of A B Both A and R are true, but R is not the correct explanation of A C A is true but R is false D A is false but R is true
Show answer & explanation

A is true but R is false

R is true but A is false: only two pairs of sides are known equal at the start, so SSS is unavailable and RHS is required.

Q89. The angle at the external point and the angle between the two radii are

Advanced
A equal B supplementary C complementary D unrelated
Show answer & explanation

supplementary

The kite has two right angles, leaving $180°$ for the other two.

Q90. If two tangents from an external point meet at 70°, the angle between the radii is

Advanced
A 70° B 110° C 20° D 35°
Show answer & explanation

110°

$180° - 70°$.

Q91. If two tangents from an external point meet at 60°, the triangle formed by the two tangents and the chord of contact is

Advanced
A equilateral B right-angled C scalene D obtuse-angled
Show answer & explanation

equilateral

$PA = PB$ makes it isosceles, and a 60° apex forces the base angles to 60° too.

Q92. A quadrilateral circumscribes a circle with sides 8, 9, 10 in order. The fourth side is

Advanced
A 9 cm B 8 cm C 11 cm D 27 cm
Show answer & explanation

9 cm

$8 + 10 = 9 + x$ gives $x = 9$.

Q93. A triangle with sides 5, 12 and 13 cm has incircle tangent lengths

Advanced
A 5, 12 and 13 cm B 2, 3 and 10 cm in that order from the 5 cm side C 6, 6.5 and 2.5 cm D 10, 3 and 2 cm
Show answer & explanation

10, 3 and 2 cm

$s = 15$, so $15-5 = 10$, $15-12 = 3$, $15-13 = 2$.

Q94. The three incircle tangent lengths of a triangle always add up to

Advanced
A the perimeter B the longest side C twice the perimeter D the semiperimeter
Show answer & explanation

the semiperimeter

$x + y + z = s$, which is how the formulas are derived.

Q95. Two tangents from $P$ touch a circle at $A$ and $B$. If $\angle APB = 90°$ and the radius is 5 cm, then $PA$ is

Advanced
A 5 cm B 10 cm C $5\sqrt2$ cm D 2.5 cm
Show answer & explanation

5 cm

$OAPB$ then has four right angles and $OA = PA$, so it is a square of side 5.

Q96. Which fact is NOT a consequence of the congruence in Theorem 2's proof?

Advanced
A $PA = PB$ B the tangents are perpendicular to each other C $OP$ bisects $\angle APB$ D $OP$ bisects $\angle AOB$
Show answer & explanation

the tangents are perpendicular to each other

The angle between the tangents depends on how far $P$ is from the centre.

Q97. As the external point moves further from the circle, the angle between the two tangents

Advanced
A gets larger B stays at 90° C gets smaller D stays the same
Show answer & explanation

gets smaller

From very far away the two tangents are almost parallel.

Q98. Every triangle has an incircle, but a quadrilateral has one only if

Advanced
A the sums of its opposite sides are equal B it is convex C it has a right angle D its diagonals are equal
Show answer & explanation

the sums of its opposite sides are equal

That is the tangential condition.

Q99. The point where the two tangents meet lies on

Advanced
A the chord of contact B the circle C a diameter through A D the perpendicular bisector of the chord of contact
Show answer & explanation

the perpendicular bisector of the chord of contact

$OP$ is that perpendicular bisector, and $P$ is on it.

Q100. Which theorem supplies the right angles in the circumscribing-quadrilateral proof?

Advanced
A neither — only Theorem 2 is needed B Theorem 1, four times C Pythagoras D the angle sum property
Show answer & explanation

neither — only Theorem 2 is needed

That proof adds four tangent-length equalities and never uses a right angle.

Q101. In a tangential quadrilateral with sides 12, 15, 18 in order, the fourth side is

Advanced
A 12 cm B 21 cm C 9 cm D 15 cm
Show answer & explanation

15 cm

$12 + 18 = 15 + x$ gives $x = 15$.

Q102. Assertion (A): A rectangle that is not a square can still have a circle touching all four of its sides. Reason (R): In a quadrilateral circumscribing a circle the sums of opposite sides are equal.

Advanced
A Both A and R are true, and R is the correct explanation of A B Both A and R are true, but R is not the correct explanation of A C A is true but R is false D A is false but R is true
Show answer & explanation

A is false but R is true

Its opposite sides give unequal sums, so no such circle exists and the assertion is false. The reason states the condition.

Q103. Assertion (A): In the figure of two tangents from an external point, the chord of contact is bisected by $OP$. Reason (R): $OAPB$ is a square, and the diagonals of a square bisect each other.

Advanced
A Both A and R are true, and R is the correct explanation of A B Both A and R are true, but R is not the correct explanation of A C A is true but R is false D A is false but R is true
Show answer & explanation

A is true but R is false

$OAPB$ is a kite, not a square - two sides are radii and two are tangents. The assertion stands, the reason is false.

Q104. A circle of radius 6 cm has an external point 10 cm from its centre. The total length of both tangents together is

Advanced
A 8 cm B 16 cm C 12 cm D 20 cm
Show answer & explanation

16 cm

Each is $\sqrt{100-36} = 8$, and there are two of them.

Q105. Prove that the two tangents drawn to a circle from one outside point have equal length.

Advanced
Show answer & explanation

—

Q106. Prove that in a quadrilateral $ABCD$ circumscribing a circle, $AB + CD = BC + DA$.

Advanced
Show answer & explanation

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Q107. Prove that a parallelogram circumscribing a circle is a rhombus.

Advanced
Show answer & explanation

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Q108. A triangle has sides 14 cm, 10 cm and 12 cm. Find the three tangent lengths of its incircle.

Advanced
Show answer & explanation

—

Q109. Explain why the angle between two tangents from an external point and the angle between the corresponding radii are supplementary.

Advanced
Show answer & explanation

—

Q110. Why can Theorem 2 not be proved by the SAS criterion?

Advanced
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Q111. (a) Prove that the two tangents drawn to a circle from one outside point have equal length. (b) List the other results that follow from the same congruence. (c) A circle of radius 9 cm has an external point 41 cm from its centre. Find each tangent length, and state the angle between each tangent and its radius.

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Q112. (a) Prove that a quadrilateral circumscribing a circle has equal sums of opposite sides. (b) Use it to prove that a parallelogram circumscribing a circle is a rhombus. (c) A quadrilateral circumscribes a circle and has consecutive sides 8 cm, 9 cm and 10 cm. Find the fourth side, and decide whether a 3 cm by 5 cm rectangle could circumscribe a circle.

Advanced
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Q113. (a) Derive the formulas for the three tangent lengths of a triangle's incircle. (b) Apply them to a triangle with sides 13 cm, 14 cm and 15 cm. (c) Explain how to check your three answers.

Advanced
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Q114. (a) Explain why the angle at the external point and the angle at the centre are supplementary. (b) Two tangents are drawn from a point $P$ to a circle of centre $O$ and radius 5 cm, and they meet at 90°. Find the tangent length and the distance $OP$. (c) Describe what happens to the angle between the tangents as $P$ moves further from the circle, and why.

Advanced
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