# Free Circles 02 Practice Test - 9th Grade

The length of the complete circle is called __ of the circle.

#### SOLUTION

Solution :

The boundary of the circle is called its circumference and the value of circumference is 2πr (where r is the radius of the circle). If we cut a circle and form a line from it, then the length of the line will be the same as the circumference of the circle.

Which of the following is a chord?

A.

Diameter

B.

C.

Segment

D.

Arc

#### SOLUTION

Solution : A

A chord is a line which touches the circle at two distinct points on its circumference.

A segment is an area cut by a chord in a circle and hence cannot be a chord.
An arc is a part of the circumference and hence cannot be a chord. The radius is not a chord as it does not touch the circle at two points on its circumference. The diameter is a chord  since it touches the circle at two points on its circumference. The line drawn through the centre of a circle to bisect a chord is __ to the chord.

#### SOLUTION

Solution : Any line passing through the centre of the circle will perpendicularly bisect the chord of the circle.

The maximum number of common tangents that can be drawn for two external touching circles is/are ___.

#### SOLUTION

Solution :

Only three tangents can be drawn passing through two external touching circles. Only one circle can be drawn through three non-collinear points.

A.

True

B.

False

#### SOLUTION

Solution : A

With three non-collinear points, only one circle can be drawn as shown below. Note the points should be non-collinear otherwise no circle can be drawn. Given a circle with O as the centre.  Find the value of x. A.

(802)

B.

(1002)

C.

50

D.

40

#### SOLUTION

Solution : A and D

ΔAOB is an isosceles triangle (OA = OB; Radii),

OAB=OBA=50.

In ΔAOB,

OAB+OBA+AOB=180

Hence, AOB=80

The angle subtended by an arc of the circle at its centre is double the angle subtended by it at any point on the remaining part of the circle.
Therefore, ACB=12AOB=40.

In the given figure, AB is the diameter of the circle. Find the value of x. A.

30

B.

45

C.

60

D.

90

#### SOLUTION

Solution : D

An angle formed by the diameter of a circle at its circumference equals 90.Hence, the value of x is  90.

What is the sum of either pair of opposite angles of a cyclic quadrilateral?

A.

90

B.

180

C.

360

D.

45

#### SOLUTION

Solution : B

The opposite angles of a cyclic quadrilateral are supplementary to each other. Hence, the sum of opposite angles in a cyclic quadrilateral is 180.

How many circles can be drawn through two points?

A.

One

B.

Two

C.

Infinite

D.

Four

#### SOLUTION

Solution : C

An infinite number of circles can be drawn passing through two different points.
The line joining these two points can be the chord of an infinite number of circles having different radius. Which of the following is/are correct?

A. The perpendicular drawn from the centre of the circle to a chord bisects the chord.
B. Congruent arcs of a circle subtend equal angles at the centre.
C. Congruent arcs of a circle subtend right angles at the centre.
D. Chords equidistant from the centre of a circle are equal in length.

#### SOLUTION

Solution : A, B, and D (i) A perpendicular drawn from the centre of the circle acts as a perpendicular bisector to the chord. (ii) Congruent arcs of a circle subtend equal angles at the centre. These angles are equal but not always a right angle. A right angle is subtended only when the arc is a semicircle. (iii) Right angles will be subtended by congruent arcs only when they are semicircles. (iv) If two chords are drawn at a same distance from the centre of a circle, they will be equal in length.