circles Flashcards
- A circle has a number of tangents equal to
(a) 0
(b) 1
(c) 2
(d) Infinite
(d)Infinite
A circle has infinitely many tangents, touching the circle at infinite points on its circumference.
- A tangent intersects the circle at:
(a) One point
(b) Two distinct point
(c) At the circle
(d) None of the above
(a)One point
A tangent touches the circle only on its boundary and do not cross through it.
- A circle can have _____parallel tangents at a single time.
(a) One
(b) Two
(c) Three
(d) Four
(b)Two
A circle can have two parallel tangents at the most.
- If the angle between two radii of a circle is 110º, then the angle between the tangents at the ends of the radii is:
(a) 90º
(b) 50º
(c) 70º
(c) 40º
(c)70º
If the angle between two radii of a circle is 110º, then the angle between tangents is 180º − 110º = 70º. (By circles and tangents properties)
- The length of the tangent from an external point A on a circle with centre O is
(a) always greater than OA
(b) equal to OA
(c) always less than OA
(d) Cannot be estimated
(c)always less than OA
Since the tangent is perpendicular to the radius of the circle, then the angle between them is 90º. Thus, OA is the hypotenuse for the right triangle OAB, which is right-angled at B. As we know, for any right triangle, the hypotenuse is the longest side. Therefore the length of the tangent from an external point is always less than the OA.
- AB is a chord of the circle and AOC is its diameter such that angle ACB = 50°. If AT is the tangent to the circle at the point A, then BAT is equal to
(a) 65°
(b) 60°
(c) 50°
(d) 40°
(c)50° ∠ABC = 90 (Angle in Semicircle) In ∆ACB ∠A + ∠B + ∠C = 180° ∠A = 180° – (90° + 50°) ∠A = 40° Or ∠OAB = 40° Therefore, ∠BAT = 90° – 40° = 50°
- If TP and TQ are the two tangents to a circle with centre O so that ∠POQ = 110°, then ∠PTQ is equal to
(a) 60°
(b) 70°
(c) 80°
(d) 90°
(b)70°
We can see, OP is the radius of the circle to the tangent PT and OQ is the radius to the tangents TQ.
So, OP ⊥ PT and TQ ⊥ OQ
∴ ∠OPT = ∠OQT = 90°
Now, in the quadrilateral POQT, we know that the sum of the interior angles is 360°
So, ∠PTQ + ∠POQ + ∠OPT + ∠OQT = 360°
Now, by putting the respective values, we get,
⇒ ∠PTQ + 90° + 110° + 90° = 360°
⇒ ∠PTQ = 70°
- The length of a tangent from a point A at a distance 5 cm from the centre of the circle is 4 cm. The radius of the circle is:
(a) 3cm
(b) 5cm
(c) 7cm
(d) 10cm
(a)3cm
AB is the tangent, drawn on the circle from point A. So, OB ⊥ AB Given, OA = 5cm and AB = 4 cm Now, In △ABO, OA2 = AB2 + BO2 (Using Pythagoras theorem) ⇒ 52 = 42 + BO2 ⇒ BO2 = 25 – 16 ⇒ BO2 = 9 ⇒ BO = 3
- If a parallelogram circumscribes a circle, then it is a:
(a) Square
(b) Rectangle
(c) Rhombus
(d) None of the above
(c)Rhombus
- Two concentric circles are of radii 5 cm and 3 cm. The length of the chord of the larger circle which touches the smaller circle is:
(a) 8
(b) 10
(c) 12
(d) 18
(a)8 From the above figure, AB is tangent to the smaller circle at point P. ∴ OP ⊥ AB By Pythagoras theorem, in triangle OPA OA2 = AP2 + OP2 ⇒ 52 = AP2 + 32 ⇒ AP2 = 25 – 9 ⇒ AP = 4 Now, as OP ⊥ AB, Since the perpendicular from the center of the circle bisects the chord, AP will be equal to PB So, AB = 2AP = 2 × 4 = 8 cm
Two concentric circles of radii a and b (a > b) are given. Find the length of the chord of the larger circle which touches the smaller circle.
a) √a^2+b^2
b) √a^2-b^2
c) 2√a^2-b^2
d) 2√a^2+b^2
c)2√a^2-b^2
- Three circles touch each other externally. The distance between their centres is 5 cm, 6 cm and 7 cm. Find the radii of the circles.
a) 2 cm, 3 cm, 4 cm
b) 1 cm, 2 cm, 4 cm
c) 1 cm, 2.5 cm, 3.5 cm
d) 3 cm, 4 cm, 1 cm
a)2 cm, 3 cm, 4 cm
- A point P is 13 cm from the centre of the circle. The length of the tangent drawn from P to the circle is 12cm. Find the radius of the circle.
a) 5cm
b) 7cm
c) 10cm
d) 12cm
a)5cm
In the adjoining figure ‘O’ is the center of circle, ∠CAO = 25° and ∠CBO = 35°. What is the value of ∠AOB?
a) 120°
b) 110°
c) 55°
d) Data insufficient
a)120°
A: What is a line called, if it meets the circle at only one point? B: Collection of all points equidistant from a fixed point is \_\_\_\_\_\_. 1: Chord 2: Tangent 3: Circle 4: Curve 5: Secant Which is correct matching? a)A-2; B-4 b)A-5; B-4 c)A-4; B-1 d)A-2; B-3
d)A-2; B-3
- A point A is 26 cm away from the centre of a circle and the length of tangent drawn from A to the circle is 24 cm. Find the radius of the circle.
a) 2√313
b) 12
c) 7
d) 10
d)10
- The quadrilateral formed by joining the angle bisectors of a cyclic quadrilateral is a
a) cyclic quadrilateral
b) parallelogram
c) square
d) Rectangle
a)cyclic quadrilateral
- ABCD is a cyclic quadrilateral PQ is a tangent at B. If ∠DBQ = 65°, then ∠BCD is
a) 35°
b) 85°
c) 90°
d) 115°
d)115°
In a circle of radius 5 cm, AB and AC are the two chords such that AB = AC = 6 cm. Find the length of the chord BC
a) None of these
b) 9.6cm
c) 10.8cm
d) 4.8cm
b)9.6cm
The distance between the centres of equal circles each of radius 3 cm is 10 cm. The length of a transverse tangent AB is
a) 10cm
b) 8cm
c) 6cm
d) 4cm
b)8cm
A point P is 10 cm from the center of a circle. The length of the tangent drawn from P to the circle is 8 cm. The radius of the circle is equal to
a) 4cm
b) 5cm
c) None of these
d) 6cm
d)6cm
In fig, O is the centre of the circle, CA is tangent at A and CB is tangent at B drawn to the circle. If ∠ACB = 75°, then ∠AOB=
a) 75°
b) 85°
c) 95°
d) 105°
d)105°
In the given figure, PAQ is the tangent. BC is the diameter of the circle. ∠BAQ = 60°, find ∠ABC :
a) 25°
b) 30°
c) 45°
d) 60°
b)30°
