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Circles Maths - Circles

CBSE - STD 10 - Maths (CBSE - English Medium). 268 questions in this chapter.

Question types

Circles question types

268 questions across 8 question groups — pick any mix to generate a Maths paper with step-by-step answer keys.

268
Questions
8
Question groups
5
Question types
Sample Questions

Circles questions

One sample from each question group in this chapter. Select any group above to see the full set with answer keys.

Q 1M.C.Q (1 Marks)1 Mark
In the given figure, QR is a common tangent to the given circles touching externally at the point T. The tangent at T meets QR at P. If PT = 3.8cm, then the length of
QR (in cm) is:
  • A
    3.8
  • ✓
    7.6
  • C
    5.7
  • D
    1.9

Answer: B.

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Q 2M.C.Q (1 Marks)1 Mark
In the figure, a quadrilateral ABCD is drawn to circumscribe a circle such that its sides AB, BC, CD and AD touch the circle at P, Q, R and S respectively. If AB = x cm,
BC= 7cm, CR = 3cm and AS = 5cm, then x =
  • A
    10
  • ✓
    9
  • C
    8
  • D
    7

Answer: B.

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Q 3M.C.Q (1 Marks)1 Mark
In the given figure, if AD, AE and BC are tangents to the circle at D, E and F respectively, Then:
  • A
    AD = AB + BC + CA
  • ✓
    2AD = AB + BC + CA
  • C
    3AD = AB + BC + CA
  • D
    4AD = AB + BC + CA

Answer: B.

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Q 4M.C.Q (1 Marks)1 Mark
The length of the tangent from a point $A$ at a circle, of radius $3 \ cm,$ is $4 \ cm.$ The distance of $A$ from the centre of the circle is:
  • A
    $\sqrt{7}\text{cm}$
  • B
    $7\ cm$
  • ✓
    $5\ cm$
  • D
    $25\ cm$

Answer: C.

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Q 5M.C.Q (1 Marks)1 Mark
If PT is tahgent drawn froth a point P to a circle touching it at T and O is the centre of the circle, then ∠OPT + ∠POT =
  • A
    30°
  • B
    60°
  • ✓
    90°
  • D
    180°

Answer: C.

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Q 6Assertion (A) & Reason (B) MCQ1 Mark
Statement-1 (A): The tangents drawn at the end points of a diameter of a circle are parallel.
Statement-2 (R): Diameter of a circle is the longest chord.
  • A
    Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
  • ✓
    Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
  • C
    Statement-1 is true, Statement- 2 is false.

Answer: B.

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Q 7Assertion (A) & Reason (B) MCQ1 Mark
Statement-1 (A): A tangent to a circle is perpendicular to the radius through the point of contact.
Statement-2 (R): The lengths of tangents drawn from an external point to a circle are equal.
  • A
    Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
  • ✓
    Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
  • C
    Statement-1 is true, Statement- 2 is false.

Answer: B.

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Q 8Assertion (A) & Reason (B) MCQ1 Mark
Statement-1 (A): In Fig. O is the centre of a circle and $P Q$ is a chord. If the tangent $P R$ at $P$ makes an angle of $50^{\circ}$ with $P Q$, then $\angle P O Q=100^{\circ}$.
Statement-2 (R): A tangent to a circle is perpendicular to the radius through the point of contact.
  • ✓
    Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
  • B
    Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
  • C
    Statement- 1 is True, Statement- 2 is False.
  • D
    Statement- 1 is False, Statement- 2 is True.

Answer: A.

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Q 9Assertion (A) & Reason (B) MCQ1 Mark
Statement-1 (A): In Fig. if AT is tangent to the circle, with centre $O$, at point $A$ such that $O T=4 cm$, and $\angle O T A=30^{\circ}$, then $A T=4 \sqrt{3} cm$.
Statement-2 (R) : A tangent to a circle is perpendicular to the radius through the point of contact.
  • A
    Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
  • B
    Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
  • C
    Statement- 1 is True, Statement- 2 is False.
  • ✓
    Statement- 1 is False, Statement- 2 is True.

Answer: D.

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Q 10Assertion (A) & Reason (B) MCQ1 Mark
Statement-1 (A): The base of an isosceles triangle is bisected at the point of contact of its incircle.
Statement-2 (R): The lengths of two tangents drawn from an external point to a circle are equal.
  • ✓
    Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
  • B
    Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
  • C
    Statement- 1 is True, Statement- 2 is False.
  • D
    Statement- 1 is False, Statement- 2 is True.

Answer: A.

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Q 222 Marks Questions2 Marks
In the figure, BOA is a diameter of a circle and the tangent at a point P meets BA produced at T. If $\angle\text{PBO}=30^{\circ}$ then find $\angle\text{PTA}.$
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Q 232 Marks Questions2 Marks
In the figure, PQL and PRM are tangents to the circle with centre O at the points Q and R respectively and S is a point on the circle such that $\angle\text{SQL}=50^{\circ}$ and
$\angle\text{SRM}=60^{\circ}.$ Then, find $\angle\text{QSR}.$
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Q 242 Marks Questions2 Marks
In the figure, CP and CQ are tangents from an external point C to a circle with centre O. AB is another tangent which touches the circle at R. If CP = 11cm and BR = 4cm, find the length of BC.
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Q 252 Marks Questions2 Marks
In the figure, PA and PB are tangents to the circle drawn from an external point P. CD is a third tangent touching the circle at Q. If PB = 10cm and CQ = 2cm, what is the length PC?
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Q 263 Marks Question3 Marks
In the given figure, there are two concentric circles with centre $O$ of radii 5cm and $ 3\ cm$. From an external point $P$, tangent $PA$ and $PB$ are drawn to these circles.
If $AP = 12\ cm$, find the length of $BP.$
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Q 293 Marks Question3 Marks
$A$ is a point at a distance $13 \ cm$ from the centre $O$ of a circle of radius $5 \ cm$ . $A P$ and $A Q$ are the tangents to the circle at $P$ and $Q$. If a tangent $B C$ is drawn at a point $R$ lying on the minor arc $P Q$ to intersect $A P$ at $B$ and $A Q$ at $C$, find the perimeter of the $\triangle A B C$.
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Q 315 Marks Questions5 Marks
In the figure, PQ is a tangent from an external point P to a circle with centre O and OP cuts the circle at T and QOP is a diameter. If $\angle\text{POR}=130^{\circ}$ and S is a point
on the circle, find $\angle1+\angle2.$
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Q 36Case study (4 Marks)4 Marks
$A$ backyard is in the shape of a triangle $A B C$ with right angle at $B, A B=7 m$ and $B C=15 m$. A circular pit was dug inside it such that it touches the walls $A C, B C$ and $A B$ at $P, Q$ and $R$ respectively such that $A P=x m$.
Based on the above information, answer the following questions:
(i) Find the length of $A R$ in terms of $x$.
(ii) Write the type of quadrilateral $B Q O R$.
(iii) (a) Find the length PC in terms of $x$ and hence find the value of $x$.
OR
(b) Find $x$ and hence find the radius $r$ of circle.

Image

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Q 37Case study (4 Marks)4 Marks
The discus throw is an event in which an athlete attempts to throw a discus. The athlete spins anti-clockwise around one and a half times through a circle, then releases the throw. When released the discus travels along tangent to the circular spin orbit.
Image
In the given figure, $A B$ is one such tangent to a circle of radius 75 cm . Point $O$ is centre of the circle and $\angle A B O=30^{\circ}$. $P Q$ is parallel to $O A$.
Based on above information find the lengths of:
(i) $A B$ $\qquad$ (ii) $O B$ $\qquad$ (iii) $A P$ $\qquad$ (iv) $P Q$
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Q 38Case study (4 Marks)4 Marks
Raghav has been selected by his school to design logo for sports day T-shirts for students and staff. The logo design is as given in the figure and he is working on the fronts and different colours according to the theme. In the given figure, a circle with centre $O$ is inscribed in a $\triangle A B C$, such that it touches the sides $A B, B C$ and $C A$ at points $D, E$ and $F$ respectively. The length of side $A B, B C$ and $C A$ are $12 cm, 8 cm$ and 10 cm respectively.
Image
(i) The length of $A D$ is
(a) 7 cm $\qquad$ (b) 8 cm $\qquad$ (c) 5 cm $\qquad$ (d) 9 cm
(ii) The length of $B E$ is
(a) 8 cm $\qquad$ (b) 5 cm $\qquad$ (c) 2 cm $\qquad$ (d) 9 cm
(iii) CF is of length
(a) 9 cm $\qquad$ (b) 5 cm $\qquad$ (c) 2 cm $\qquad$ d) 3 cm
(iv) If radius of the circle is 4 cm , then area of $\triangle O A B$ is
(a) 20 cm $\qquad$ (b) 36 cm $\qquad$ (c) 24 cm $\qquad$ (d) 48 cm
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Q 39Case study (4 Marks)4 Marks
London eye is an amusement ride consisting of a rotating upright big wheel with multiple passenger-carrying components (commonly referred to as passenger cars, cabins, tubs, capsules, gondolas or pods) attached to the rim in such a way that as the wheel turns, they are kept upright usually by gravity. After taking a ride in London eye, Anu came out from the crowd and was observing her friends who were enjoying the ride. She was curious about the different angles and measures that the wheel will form. She makes a figure as given below.
Image
(i) In Fig. the measure of $\angle R O Q$ is
(a) $60^{\circ}$ $\qquad$ (b) $100^{\circ}$ $\qquad$ (c) $150^{\circ}$ $\qquad$ (d) $90^{\circ}$
(ii) In Fig. 8.33, The measure of $\angle R Q P$ is
(a) $75^{\circ}$ $\qquad$ (b) $60^{\circ}$ $\qquad$ (c) $30^{\circ}$ $\qquad$ (d) $90^{\circ}$
(iii) In Fig. 8.33, the measure of $\angle R S Q$ is
(a) $60^{\circ}$ $\qquad$ (b) $75^{\circ}$ $\qquad$ (c) $100^{\circ}$ $\qquad$ (d) $30^{\circ}$
(iv) In Fig. 8.33, the measure of $\angle O R P$ is
(a) $90^{\circ}$ $\qquad$ (b) $70^{\circ}$ $\qquad$ (c) $100^{\circ}$ $\qquad$ (d) $60^{\circ}$
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Q 40Case study (4 Marks)4 Marks
The chain and gears of bicycles or motorcycles or belt around pulleys are some real-life illustrations of tangents to circles.
Image
(i) $P I$ and $P A$ are tangents to the circle from point $P$. If arc $I Z A$ subtends an angle $240^{\circ}$ at the centre of the circle, then $\angle I P A=$
(a) $120^{\circ}$ $\qquad$ (b) $90^{\circ}$ $\qquad$ (c) $60^{\circ}$ $\qquad$ (d) $30^{\circ}$
(ii) If $I P=15 cm$, then $A I=$
(a) 7.5 cm $\qquad$ (b) 15 cm $\qquad$ (c) 30 cm $\qquad$ (d) 18 cm
(iii) If $I P=21 cm$ and measure of $A P$ is $x^2+5$, then $x=$
(a) 4 $\qquad$ (b) 16 $\qquad$ (c) $\sqrt{26}$ $\qquad$ (d) $\sqrt{30}$
(iv) $\angle O I P+\angle A P O=$
(a) $90^{\circ}$ $\qquad$ (b) $60^{\circ}$ $\qquad$ (c) $120^{\circ}$ $\qquad$ (d) $150^{\circ}$
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