Circles Test

Result

Circles Test - 16

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Weekly Quiz Competition

Question 1
1 / -0

In the given figure, O is the centre of the circle and PQ is a chord.

Which of the following relations is true?

A
PQ = 2MQ

B
MQ = PQ

C
OM = MQ

D
OM = PQ

Solution

Angular bisector divides the chord into equal parts.
since OP = OQ (radius of a circle)
So, PM = MQ
∴ PQ = PM + MQ
PQ = 2 MQ
Therefore, option A is correct.
Question 2
1 / -0

In the given figure, O is the centre of the circle, RPM = 30° and TPN = 60°. Then find the measure of PMN.

A
30°

B
60°

C
45°

D
90°

Solution

From the figure, given in the question we get that
RPT $∥$ MON
So, $∠ T P N = ∠ P M N$ (alternate angles)
$∠ T P N = 30 °$ (given)
∴ $∠ P M N = 30 °$
Therefore, option B is correct.
Question 3
1 / -0

In the given figure, PB is the tangent to a circle with centre A. If AB = 5 cm and PA = 8 cm, then find the length of PB.

A
7 cm

B
8 cm

C
cm

D
cm

Solution

From the figure we get that
$A B = 5 c m$ (Base)
$P A = 8 c m$ (Hypotenuse)
$P B$ is a Perpendicular.
So, we will apply the Pythagoras Theorem.
${(P A)}^{2} = {(A B)}^{2} + {(P B)}^{2}$
${(8)}^{2} = {(5)}^{2} + {(P B)}^{2}$
${(P B)}^{2} = {(8)}^{2} - {(5)}^{2}$
${(P B)}^{2} = 64 - 25 = 39$
$P B = \sqrt{39} c m$
Therefore, option D is correct.
Question 4
1 / -0

In the given figure, the lines PT and PAB are

A
both tangents

B
both secants

C
a secant and a chord, respectively

D
a tangent and a secant, respectively

Solution

A tangent to a circle is a straight line that touches the circle at only one point. It is perpendicular to the radius at the point of tangency.
So, PT is a tangent in the figure.
A secant of a curve is a line that intersects the curve at a minimum of two distinct points.
So, PAB is secant in the figure.
Therefore, option D is correct.
Question 5
1 / -0

In the given figure, O is the centre of the circle. If ACB = 55°, find ADB.

A
110°

B
35°

C
60°

D
55°

Solution

An angle that is made on the semi-circle angle, the corresponding angle will be the same.
When two lines are crossed by another line the angles in matching corners are called corresponding angles. When the two lines are parallel corresponding angles are equal.
$∠ A C B = 55 °$ (given)
∴ $∠ A C B = ∠ A D B$ (corresponding angles)
$∠ A D B = 55 °$
Therefore, option D is correct.
Question 6
1 / -0

In the given figure, O is the centre and AB is any chord. If AM = MB, which of the following relations is true?

A
OM = MB

B
OM $⊥$ AB

C
OM = AM

D
$∠$ OBM = $∠$ AOM

Solution

The line from the center of a circle bisects the chord equally.
OA = OB (radius of a circle)
it means that line is perpendicular on the chord.
AM = MB (given)
∴ $O M ⊥ A B$
Therefore, option B is correct.
Question 7
1 / -0

In the given figure, O is the centre of the circle. A and B are two points on the circle.

Which of the following statements is definitely true?

A
$∆ O A B$ is equilateral.

B
$∆ O A B$ is a right triangle.

C
$∆ O A B$ is isosceles.

D
$∆ O A B$ is scalene.

Solution

The distance from the center to the edge of the circle is always the same.
Points A and B of the same distance from the point O that is at the center of the circle.
So, OA = OB
$∠ O A B = ∠ O B A$ (angles opposite to equal sides are equal)
So, we get that $△ O A B$ is an isosceles triangle.
Therefore, option C is correct.
Question 8
1 / -0

In the given figure, AB and AC are two tangents to a circle with centre O.

What is the perimeter of the figure OCAB, if AB = 7 cm and OC = 3 cm?

A
10 cm

B
40 cm

C
14 cm

D
20 cm

Solution

If two tangent segments are drawn to one circle from the same external point, then they are congruent.
AB = 7 cm (given)
OC = 3 cm (given)
So, AB = AC = 7 cm [Tangent segments drawn to the same circle from the same point (there are two for every circle) are equal]
and OC = OB = 3 cm [Radius of circle]
∴ the perimeter of OCAB = OC + AC + AB + OB
= 3 + 7 + 7 + 3 = 20 cm
Therefore, option D is correct.
Question 9
1 / -0

In the given figure, $∠$ BAT = x. If BC || AT and O is the centre of the circle, then find the measure of $∠$ ACB.

A
90^o- x

B
x

C
60^o

D
180^o - x

Solution

From the figure, given in the question we get that
$B C ∥ A T$
So, $∠ B A T = ∠ A C B$ (alternate angles)
$∠ B A T = x$ (given)
∴ $∠ A C B = x$
Therefore, option B is correct.
Question 10
1 / -0

In the given figure, O is the centre. P, Q and R are three points on the circle.

Which of the following relations is true?

A
OP > OQ

B
OP + OQ = OR

C
2OR = OP + OQ

D
OP - OQ = OR

Solution

The distance from the center to the edge of the circle is always the same.
Points P, Q, and R of the same distance from the point O that is at the center of the circle.
∴ length of OR, OQ, and OP will be the same, as they represent radius of the circle.
i.e., OR = OQ = OP= radius of circle
So, we can write this as 2OR = OP + OQ.
Therefore, option C is correct.