KSEEB Solutions for Class 10 Maths Chapter 4 Circles Additional Questions

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Karnataka State Syllabus Class 10 Maths Chapter 4 Circles Additional Questions

I. Multiple Choice Questions:

Question 1.
Line segment joining the centre and a point on the circle is called
(a) radius
(b) diameter
(c) Chord
(d) Arc
Answer:
(a) radius

Question 2.
Part of a circle is called
(a) Chord
(b) diameter
(c) Segment
(d) Arc
Answer:
(d) Arc

Question 3.
The biggest chord in a circle is called
(a) radius
(b) diameter
(c) chord
(d) Arc
Answer:
(b) diameter

Question 4.
The region, bounded by a major arc and a chord is called
(a) Segment
(b) major segment
(c) minor segment
(d) major arc
Answer:
(b) major segment

Question 5.
The length of the biggest chord is 8 cm then the value of radius is
(a) 8 cm
(b) 4 cm
(c) 3 cm
(d) 5 cm
Answer:
(b) 4 cm

Question 6.
How many radius can be drawn in circle
(a) 1
(b) 2
(c) only 3
(d) many
Answer:
(d) many

Question 7.
An angle in a semicircle is.
(a) 60°
(b) 30°
(c) 90°
(d) 180°
Answer:
(c) 90°

Question 8.
Equal chords of a circle are.
(a) Equidistant from the centre.
(b) Equal
(c) Unequal
(d) Not equidistant from the centre
Answer:
(b) Equal

Question 9.
If the length of the chord increases its perpendicular distance from the centre.
(a) Increases
(b) Decreases
(c) Equal
(d) Constant
Answer:
(b) Decreases

Question 10.
The perpendicular distance between the biggest chord and the centre is.
(a) zero
(b) Equal
(c) 9 cm
(d) 10cm
Answer:
(a) zero

Question 11.
In a circle angles in the major segment are called.
(a) Obtuse angles
(b) Acute angles.
(c) Right angles
(d) Complete angle
Answer:
(b) Acute angles.

Question 12.
In a circle angles in the minor segment are called.
(a) Obtuse angles
(b) Acute angles.
(c) Right angles
(d) zero angle
Answer:
(a) Obtuse angles

Question 13.
In a circle angles in the same segment are
(a) Not equal
(b) Right angles
(c) Equal
(d) zero angle.
Answer:
(c) Equal

Question 14.
Circles having the same centre but different radii are called.
(a) Congruent circles
(b) Concentric circles
(c) Equal circles
(d) None of these
Answer:
(b) Concentric circles

Question 15.
Circles having same radii but different centres are called
(a) Congruent circles
(b) Concentric circles
(c) Equal circles.
(d) Intersecting circles
Answer:
(a) Congruent circles

Question 16.
The number of circles are drawn through three non-collinear points in a plane is.
(a) 1
(b) 2
(c) 3
(d) 4
Answer:
(a) 1

Question 17.
A line which intersects a circle in two points is called
(a) A secant
(b) A chord
(c) An arc
(d) A tangent
Answer:
(a) A secant

Question 18.
A line which intersects a circle in only one point is called
(a) A secant
(b) A tangent
(c) A chord
(d) A diameter
Answer:
(b) A tangent

Question 19.
A tangent to a circle intersects the circle is
(a) one point only
(b) Two points
(c) No point
(d) Three points
Answer:
(a) one point only

Question 20.
A secant of a circle intersects the circle in
(a) only one point
(b) Two points
(c) Three points
(d) No point
Answer:
(b) Two points

Question 21.
The point where a tangent line intersects a circle is called the
(a) centre
(b) point of contact
(c) End-point
(d) None of these
Answer:
(b) point of contact

Question 22.
The angle between the tangent at any point of a circle and the radius through the point of contact is
(a) 60°
(b) 90°
(c) 45°
(d) 30°
Answer:
(b) 90°

Question 23.
How many tangents can be drawn to a circle at any point of it?
(a) 1
(b) 2
(c) 3
(d) None of these
Answer:
(a) 1

Question 24.
How many parallel tangents can a circle have at the most?
(a) 1
(b) 2
(c) 4
(d) 3
Answer:
(b) 2

Question 25.
Two circles of radii 5 cm and 3 cm touch each other externally. The distance between their centres is
(a) 5 cm
(b) 3 cm
(c) 2 cm
(d) 8 cm
Answer:
(d) 8 cm

Question 26.
Two circles of radii 5 cm and 3 cm touch each other internally. The distance between their centres is
(a) 5 cm
(b) 3 cm
(c) 2 cm
(d) 8 cm
Answer:
(c) 2 cm

Question 27.
The tangents at the endpoints of a diameter of circle are
(a) perpendicular
(b) parallel
(c) intersecting
(d) inclined at 60°
Answer:
(b) parallel

Question 28.
The length of the tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. The radius of the circle is
(a) 3 cm
(b) 2 cm
(c) 5 cm
(d) 4 cm
Answer:
(a) 3 cm

Question 29.
Two concentric circles of radii 5 cm, and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
(a) 8 cm
(b) 6 cm
(c) 4 cm
(d) 10 cm
Answer:
(a) 8 cm

Question 30.
How many tangent lines can be drawn to a circle from a point outside the circle?
(a) 1
(b) 2
(c) 3
(d) 4
Answer:
(b) 2

Question 31.
How many tangents can be drawn to a circle have
(a) 2
(b) infinitely many
(c) one
(d) no
Answer:
(b) infinitely many

Question 32.
In the following figure, |PBA is
(a) 60°
(b) 30°
(d) 45°
(d) none of these

Answer:
(a) 60°

Question 33.
In the following figure, Find AP if AB = 5 cm
(a) 5 cm
(b) 4 cm
(c) 3 ccm
(d) 2 cm

Answer:
(a) 5 cm

Question 34.
In the following figure, find the perimeter of A APQ if AB = 6 cm.
(a) 10 cm
(b)12cm
(c) 15 cm
(d) 6 cm

Answer:
(b)12cm

Question 35.
In the following Figure, find the length of the chord AB if PA = 4 cm and OP = 3 cm.
(a) 2 cm
(b) 4 cm
(c) 10cm
(d) 5 cm

Answer:
(c) 10cm

II. Short Answer Questions:

Question 1.
Define concentric circles.
Answer:
Circles which are having same centre and different radii are called concentric circles.

Question 2.
Define congruent circles.
Answer:
Circles which are having same radii & different centres are called congruent circles.

Question 3.
Define the sector of a circle.
Answer:
The region bounded by an arc of a circle and its two bounding radii is called sector.

Question 4.
Name the biggest chord of a circle.
Answer:
Diameter

Question 5.
Write the formula to find the perimeter of a circle.
Answer:
Circumference = C = 2nx.

Question 6.
Name the angle formed in a semi-circle is
Answer:
Right angle

Question 7.
Name the angle formed in a major segment.
Answer:
Acute angle

Question 8.
Name the angle formed in a minor segment.
Answer:
Obtuse angle.

Question 9.
The length of the tangent to a circle from a point P, which is 25 cm away from the centre, is 24 cm. What is the radius of the circle?
Answer:

Question 10.
In a quadrilateral, ABCD circumscribes a circle with centre O. If ∠AOB = 110°, then find ∠COD.
Answer:

Question 11.
Write the relationship between radius and diameter.
Answer:
d = 2r (or) r = d/2

Question 12.
Angles formed in a same segment are ________
Answer:
Equal

III. Long Answer Questions:

Question 1.
In the following figure if ∠DAB = 60°
and ∠ACB = 70°, find the measure of ∠DBA.
Answer:

Question 2.
Prove that the perpendicular from the centre of a circle to a chord, bisect the chord.
Answer:

AB is the chord of a circle with centre O and OD ⊥ AB. We have to prove that AD = DB
In ∆ ODA and ∆ ODB.
OA = OB (radii)
OD = OD (common)
∠ODA = ∠ODB = 90° (OD ⊥ AB)
∆ ODA = ∆ ODB (R H S)
AD = DB [C.P.C.T].

Question 3.
If two equal chords of a circle intersect within the circle, prove that the segments of an chord are equal to corresponding segment of the other chord.
Answer:

OL ⊥ AB and OM ⊥ CD are drawn and OP is joined.
In ∆ OLP and ∆ OMP, since equal chords are at equal distance from centres.
OL = OM (OP common)
∠OLP = ∠OMP =90°
∆ OPL ≅ ∆ OPM. (R H S)
=> PL = PM [C.P.C.T]
But, AL = CM
=> AL – PL = CM – PM
=> AP = CP
Also, AB – AP = CD – CP
∴BP = DP
Thus, corresponding segments are equals.

Question 4.
If two interesting chords of a circle make equal angel with that: diameter passing through their point of intersecting. Prove that the chords are equal.
Answer:

OM ⊥ AB and ON⊥CD
In ∆ OME and ∆ ONE

OE = OE (common)
∴ By A AS congruence
∆ OME ≅ ∆ ONE
OM = ON (C.P.C.T)
∴ AB = GD [Distance between the center to the chords equal.]
∴ Length of the chords are equal.

Question 5.
Two concentric circles of radii 13 cm and 5 cm are drawn. Find the length of the chord of the outer circle which touches the inner circle.
Answer:

In A OPB [P = 90°
OB2 = OP2 + BP2
(13)2 = (5)2 + BP2
BP2 = 165 – 25 = 144

AB = 2BP = 2 x 12 = 24cm

Question 6.
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ = 2∠OPQ
Answer:

‘O’ is the center of a circle.
AP & AQ are tangents.
In A POQ, OP = OQ [radius]

Question 7.
Circles C₁ and C₂ touch internally at a point A and AB is a chord at the circle C₁ intersecting C₂ at P. Prove that AP = PB.
Answer:

Join OA, OP and OB.

HYP OA = HYP OB [radii]
OP = OP [common]
∆ OAP ≅ ∆ OPB [by RHS theorem]
AP = PB [C.P.C.T]

Question 8.
A circle is touching the side BC of ∆ ABC at P. AB and AC when produced are touching the circle at Q and R respectively. Prove that AQ = V₂ (Perimeter of ∆ ABC)
Answer:

AQ = AR → (1) [tangents drawn from A]
BP = BQ → (2) [tangents drawn from B]
CQ = CR → (3) [tangents drawn from C]
Perimeter of ∆ ABC = AB + BC + AC
= AB + BP + PC + AC
= AB + BQ + CR + AC
[From (1), (2) & (3)]
= AQ + AR
= AQ + AQ [∴ using(1)]
Perimeter of ∆ ABC = 2 AQ
AQ = 1/2 (Perimeter of ∆ ABC)

Question 9.
A straight line drawn through the point of contact of two circles with centres A and B intersect the circles at P and Q respectively. Show that AP and BQ are parallel.
Answer:

In ∆ APC
AP = AC [radii of same circle]

Question 10.
Two circles with centres X and Y touch each other externally at P. Two diameters AB and CD are drawn one in each circle parallel to other. Prove that B, P and C are coliinear.
Answer:

∴ APD and BPC are two intersecting lines
∴ BPC is a straight line
∴ B, P, C are coliinear.