In a circle, a chord $PQ$ is 16 cm long and is perpendicular to the radius $OP$ at $P$. If $OP = 8 \, \text{cm}$, what is the distance from the center $O$ to the midpoint of $PQ$?

Correct Option: A. Solution: Since $PQ$ is perpendicular to $OP$ at $P$, the midpoint of $PQ$ is at $P$. Thus, the distance from $O$ to the midpoint of $PQ$ is the same as $OP$, which is $8 \, \text{cm}$.

Circles

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Summary

Chapter Summary: Circles

Key Points

A circle is defined as a collection of all points in a plane at a constant distance (radius) from a fixed point (centre).
Types of Lines with Respect to a Circle:
- Non-Intersecting Line: No common points with the circle.
- Secant: Two common points with the circle.
- Tangent: One common point with the circle.

Important Properties

The tangent to a circle is perpendicular to the radius at the point of contact.
The lengths of the two tangents drawn from an external point to a circle are equal.

Summary of Activities

Tangent Existence:
- No tangent can be drawn from a point inside the circle.
- One tangent can be drawn from a point on the circle.
- Two tangents can be drawn from a point outside the circle.

Theorems

Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Exercises

Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.
Find the radius of a circle given the length of a tangent and the distance from the centre.
Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

Learning Objectives

Understand the concept of tangents to a circle.
Identify the properties of tangents, including their relationship with the radius.
Prove that the lengths of tangents from an external point to a circle are equal.
Demonstrate the number of tangents that can be drawn from various positions relative to a circle.
Apply theorems related to tangents and circles in problem-solving.

Detailed Notes

Chapter 10: Circles

10.1 Introduction

A circle is a collection of all points in a plane at a constant distance (radius) from a fixed point (centre).
Key terms related to circles: chord, segment, sector, arc.

10.2 Tangent to a Circle

A tangent intersects the circle at only one point.
Activities to Understand Tangents:
- Activity 1: Rotate a straight wire around a point on a circular wire to observe tangent behavior.
- Case 1: No tangent exists for points inside the circle.
- Case 2: One tangent exists for points on the circle.
- Case 3: Two tangents exist for points outside the circle.

10.3 Number of Tangents from a Point on a Circle

The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Theorem 10.1: The tangent at a point P of a circle is perpendicular to the radius OP.

10.4 Summary

The meaning of a tangent to a circle.
The tangent to a circle is perpendicular to the radius through the point of contact.
The lengths of the two tangents from an external point to a circle are equal.

Important Diagrams

Fig. 10.1: Circle and Line Interactions

(i) Non-intersecting line.
(ii) Secant intersecting at two points.
(iii) Tangent intersecting at one point.

Fig. 10.10: Tangents and Chords

Circle with center O, tangent TP, secant PQ, and radius OP.

Fig. 10.12: Circle Inscribed in a Quadrilateral

Points of tangency labeled R, Q, P, S.

Fig. 10.14: Triangle with Inscribed Circle

Triangle ABC with incircle centered at O, radius extending to point D.

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips

Common Pitfalls

Misunderstanding Tangents: Students often confuse tangents with secants. Remember, a tangent touches the circle at exactly one point, while a secant intersects it at two points.
Drawing Tangents from Points: When asked to draw tangents from a point inside the circle, students may mistakenly think they can draw one. There are no tangents from a point inside the circle.
Identifying Points of Contact: Students may forget to label the points where tangents touch the circle, which is crucial for clarity in geometric proofs.

Tips for Success

Visualize the Concepts: Use diagrams to understand the relationships between tangents, secants, and circles. Drawing can help clarify how many tangents can be drawn from a point relative to the circle's position.
Practice Theorems: Familiarize yourself with key theorems, such as the tangent being perpendicular to the radius at the point of contact. This understanding is essential for solving related problems.
Check Your Work: When calculating lengths of tangents, ensure you apply the Pythagorean theorem correctly, especially when dealing with right triangles formed by the radius and tangent.
Review Common Questions: Be prepared for questions that ask about the number of tangents from various points (inside, on, and outside the circle) and the properties of tangents related to angles and lengths.

Practice & Assessment

Multiple Choice Questions

They are parallel

They are perpendicular

They form an acute angle

They form an obtuse angle

Correct Answer: B

Solution:

The tangent to a circle is perpendicular to the radius through the point of contact.

The tangents are of different lengths.

The tangents are equal in length.

The tangents are perpendicular to each other.

The tangents are parallel to each other.

Correct Answer: B

Solution:

The lengths of the two tangents from an external point to a circle are equal.

7, \text{cm}

8, \text{cm}

9, \text{cm}

10, \text{cm}

Correct Answer: C

Solution:

Let

M

be the midpoint of the chord

AB

. Then

AM = MB = \frac{12}{2} = 6\, \text{cm}

. In the right triangle

OMA

OM^2 + AM^2 = OA^2

. Substituting the given values,

5^2 + 6^2 = OA^2

. Solving for

OA

, we get

OA = \sqrt{5^2 + 6^2} = \sqrt{25 + 36} = \sqrt{61} \approx 7.81\, \text{cm}

10 cm

11 cm

9 cm

12 cm

Correct Answer: B

Solution:

Let

M

be the midpoint of

AB

. Then

OM \perp AB

and

AM = MB = 7\, \text{cm}

. By the Pythagorean theorem in

\triangle OMA

OA^2 = OM^2 + AM^2 = 6^2 + 7^2 = 36 + 49 = 85

. Thus, the radius

OA = \sqrt{85} \approx 11\, \text{cm}

Secant

Tangent

Chord

Radius

Correct Answer: B

Solution:

A tangent is a line that intersects a circle at exactly one point.

Tangent

Secant

Non-intersecting line

Chord

Correct Answer: C

Solution:

A non-intersecting line is a line that does not intersect the circle at any point.

It intersects the circle at two points.

It is a line that never touches the circle.

It touches the circle at exactly one point.

It is the longest line that can be drawn inside the circle.

Correct Answer: C

Solution:

A tangent to a circle touches the circle at exactly one point.

Tangent

Secant

Chord

Diameter

Correct Answer: C

Solution:

A line segment with both endpoints on a circle is called a chord.

The lengths are equal.

The lengths are different.

The lengths are proportional to the radius.

The lengths depend on the angle between them.

Correct Answer: A

Solution:

The lengths of the two tangents from an external point to a circle are equal.

Diameter

Chord

Radius

Tangent

Correct Answer: C

Solution:

The length from the center of a circle to any point on the circle is called the radius.

Tangent

Secant

Chord

Radius

Correct Answer: C

Solution:

A chord is a line segment with both endpoints on the circle.

24 cm

18 cm

20 cm

21 cm

Correct Answer: A

Solution:

Using the Pythagorean theorem in the right triangle

OTP

, where

OT

is the hypotenuse,

OP

is one leg, and

TP

is the other leg, we have

OT^2 = OP^2 + TP^2

. Therefore,

TP = \sqrt{OT^2 - OP^2} = \sqrt{25^2 - 7^2} = \sqrt{625 - 49} = \sqrt{576} = 24

cm.

18 cm

22 cm

20 cm

16 cm

Correct Answer: A

Solution:

In a quadrilateral with an inscribed circle, the sum of the lengths of opposite sides is equal. Thus,

AB + CD = BC + DA

. Therefore,

8 + 12 = 10 + DA \Rightarrow DA = 18\, \text{cm}

The tangent is parallel to the radius.

The tangent is perpendicular to the radius.

The tangent bisects the radius.

The tangent is equal in length to the radius.

Correct Answer: B

Solution:

The tangent to a circle is perpendicular to the radius through the point of contact.

0 degrees

45 degrees

90 degrees

180 degrees

Correct Answer: C

Solution:

The tangent to a circle is perpendicular to the radius through the point of contact, forming a 90-degree angle.

11, \text{cm}

12, \text{cm}

13, \text{cm}

14, \text{cm}

Correct Answer: B

Solution:

The length of

BC

is the sum of the segments

BD

and

DC

. Therefore,

BC = BD + DC = 7\, \text{cm} + 5\, \text{cm} = 12\, \text{cm}

Zero points

One point

Two points

Three points

Correct Answer: C

Solution:

A secant of a circle intersects the circle at two points.

They are unequal

They are equal

One is double the other

They are parallel

Correct Answer: B

Solution:

The lengths of the two tangents from an external point to a circle are equal.

A line that intersects the circle at exactly one point.

A line that intersects the circle at exactly two points.

A line that does not intersect the circle at all.

A line that is perpendicular to the radius at the point of contact.

Correct Answer: B

Solution:

A secant is a line that intersects the circle at exactly two points.

Tangent

Secant

Chord

Diameter

Correct Answer: B

Solution:

A line that intersects a circle at two distinct points is called a secant.

12 cm

10 cm

8 cm

15 cm

Correct Answer: A

Solution:

Using the Pythagorean theorem in the right triangle

OTP

OT^2 = OP^2 + PT^2

. So,

13^2 = 5^2 + PT^2

, which simplifies to

PT = 12 \, \text{cm}

A collection of points in a plane equidistant from a fixed point.

A polygon with four sides.

A line that intersects a plane at one point.

A three-dimensional shape with no edges.

Correct Answer: A

Solution:

A circle is defined as the set of all points in a plane that are at a constant distance (radius) from a fixed point (center).

4 cm

6 cm

8 cm

10 cm

Correct Answer: A

Solution:

Since

PQ

is perpendicular to

OP

P

, the midpoint of

PQ

is at

P

. Thus, the distance from

O

to the midpoint of

PQ

is the same as

OP

, which is

8 \, \text{cm}

It touches the circle at exactly one point.

It is perpendicular to the radius at the point of contact.

It can intersect the circle at two points.

The lengths of tangents from an external point are equal.

Correct Answer: C

Solution:

A tangent cannot intersect the circle at two points; that is a property of a secant.

Tangent

Chord

Secant

Diameter

Correct Answer: C

Solution:

A secant is a line that intersects a circle at two distinct points.

3, \text{cm}

4, \text{cm}

5, \text{cm}

6, \text{cm}

Correct Answer: B

Solution:

Since

O

is the center of the circle and

AB

is a chord, the perpendicular from

O

AB

bisects

AB

. Let

M

be the midpoint of

AB

. Then

AM = MB = \frac{8}{2} = 4\, \text{cm}

. In the right triangle

OAM

OM^2 + AM^2 = OA^2

. Substituting the given values,

OM^2 + 4^2 = 5^2

. Solving for

OM

, we get

OM = \sqrt{5^2 - 4^2} = \sqrt{25 - 16} = \sqrt{9} = 3\, \text{cm}

Tangent

Secant

Chord

Radius

Correct Answer: B

Solution:

A line that intersects a circle at two distinct points is called a secant.

One

Two

Three

Infinitely many

Correct Answer: B

Solution:

From an external point, exactly two tangents can be drawn to a circle.

10 cm

12 cm

14 cm

16 cm

Correct Answer: B

Solution:

The length of the chord

PQ

can be calculated using the Pythagorean theorem in the right triangle

OPQ

. Since

OP = 6

cm and

OQ = 8

cm, the length of

PQ

is given by

PQ = \sqrt{OQ^2 - OP^2} = \sqrt{8^2 - 6^2} = \sqrt{64 - 36} = \sqrt{28} = 12

cm.

12 cm

10 cm

8 cm

15 cm

Correct Answer: A

Solution:

By the Pythagorean theorem in the right triangle

OTP

OT^2 = OP^2 + TP^2

. Substituting the given values:

13^2 = 5^2 + TP^2

, we get

169 = 25 + TP^2

. Solving for

TP

, we find

TP = 12\, \text{cm}

13 cm

12 cm

5 cm

17 cm

Correct Answer: A

Solution:

Since

PQ

is a tangent at

P

OP

is perpendicular to

PQ

. By the Pythagorean theorem,

OQ^2 = OP^2 + PQ^2 = 5^2 + 12^2 = 25 + 144 = 169

. Therefore,

OQ = \sqrt{169} = 13\, \text{cm}

84 cm²

90 cm²

96 cm²

100 cm²

Correct Answer: A

Solution:

The semi-perimeter

s

of the triangle is

\frac{13 + 14 + 15}{2} = 21 \text{ cm}

. Using Heron's formula, the area

A

of the triangle is

\sqrt{s(s-a)(s-b)(s-c)} = \sqrt{21(21-13)(21-14)(21-15)} = \sqrt{21 \times 8 \times 7 \times 6} = 84 \text{ cm}^2

Tangent

Secant

Chord

Diameter

Correct Answer: B

Solution:

A line that intersects a circle at two distinct points is called a secant.

10 cm

12 cm

15 cm

20 cm

Correct Answer: B

Solution:

The perpendicular from the center

O

to the chord

AB

bisects the chord. Thus,

OM = 8\, \text{cm}

and

AM = 8\, \text{cm}

. Using the Pythagorean theorem in

\triangle OMA

OA^2 = OM^2 + AM^2 = 8^2 + 8^2 = 64 + 64 = 128

. Therefore,

OA = \sqrt{128} = 8\sqrt{2} \approx 11.31\, \text{cm}

, which rounds to

12\, \text{cm}

A line that intersects the circle at two points.

A line that does not intersect the circle.

A line that intersects the circle at exactly one point.

A line that passes through the center of the circle.

Correct Answer: C

Solution:

A tangent to a circle is a line that intersects the circle at exactly one point.

They are of different lengths.

They are perpendicular to each other.

They are equal in length.

They pass through the center of the circle.

Correct Answer: C

Solution:

The lengths of the two tangents from an external point to a circle are equal.

A line that touches the circle at one point.

A line that intersects the circle at two points.

A line segment with both endpoints on the circle.

A line that does not intersect the circle.

Correct Answer: C

Solution:

A chord is a line segment with both endpoints on the circle.

Tangent

Secant

Non-intersecting line

Chord

Correct Answer: C

Solution:

A non-intersecting line is a line that does not intersect a circle at any point.

12 cm

13 cm

5 cm

8 cm

Correct Answer: A

Solution:

Using the Pythagorean theorem in the right triangle

OTP

, where

OP

is the radius and

OT

is the hypotenuse, we have

OT^2 = OP^2 + TP^2

. Substituting the known values:

13^2 = 5^2 + TP^2

, which simplifies to

169 = 25 + TP^2

. Solving for

TP

, we get

TP^2 = 144

, hence

TP = 12 \text{ cm}

6 cm

7 cm

8 cm

9 cm

Correct Answer: B

Solution:

The length of

AB

is the sum of the segments

AR

and

RB

, which is

4 \, \text{cm} + 3 \, \text{cm} = 7 \, \text{cm}

Chord

Diameter

Radius

Secant

Correct Answer: C

Solution:

A line drawn from the center of the circle to a point on the circle is called a radius.

12 cm

14 cm

16 cm

18 cm

Correct Answer: B

Solution:

Using the power of a point theorem, the length of the chord

AB

can be calculated using the formula

AB = \sqrt{2 \times (r^2 - (\frac{OA + OB}{2})^2)}

. Substituting the given values:

AB = \sqrt{2 \times (10^2 - (\frac{7 + 9}{2})^2)} = \sqrt{2 \times (100 - 64)} = \sqrt{72} = 14 \text{ cm}

10 cm

15 cm

20 cm

5 cm

Correct Answer: A

Solution:

The lengths of tangents drawn from an external point to a circle are equal. Therefore,

TP = TQ = 10\, \text{cm}

36 cm

42 cm

44 cm

48 cm

Correct Answer: C

Solution:

The perimeter of the quadrilateral

ABCD

is the sum of the sides

AB

BC

CD

, and

DA

. Since the circle is inscribed,

AP = AS

BQ = BP

CR = CQ

, and

DS = DR

. Therefore, the perimeter is

AB + BC + CD + DA = (AP + BP) + (BQ + CQ) + (CR + DR) + (DS + AS) = (3 + 4) + (4 + 5) + (5 + 6) + (6 + 3) = 7 + 9 + 11 + 9 = 44

cm.

Tangent

Secant

Non-intersecting line

Chord

Correct Answer: C

Solution:

A line that does not intersect a circle at any point is called a non-intersecting line.

14 cm

12 cm

16 cm

10 cm

Correct Answer: A

Solution:

The length of the side

BC

is the sum of the segments

BD

and

DC

. Therefore,

BC = BD + DC = 8 + 6 = 14\, \text{cm}

Tangent

Secant

Chord

Diameter

Correct Answer: B

Solution:

A line that intersects a circle at two distinct points is called a secant.

4 cm

5 cm

6 cm

7 cm

Correct Answer: C

Solution:

Since

A

and

B

are points on the circle,

OA

and

OB

are radii of the circle. Given

OA = OB = 5\, \text{cm}

, and

AB = 6\, \text{cm}

, the length of the chord

AB

is indeed

6\, \text{cm}

as given.

A tangent is parallel to the radius of the circle.

A tangent is perpendicular to the radius of the circle at the point of contact.

A tangent is a line that passes through the center of the circle.

A tangent is a line that intersects the circle at two points.

Correct Answer: B

Solution:

The tangent to a circle is perpendicular to the radius through the point of contact.

12 cm

9 cm

6 cm

15 cm

Correct Answer: A

Solution:

Using the Pythagorean theorem in the right triangle

OTP

, where

TP

is the tangent and

OT

is the hypotenuse, we have

OP = \sqrt{OT^2 - TP^2} = \sqrt{15^2 - 9^2} = \sqrt{225 - 81} = \sqrt{144} = 12

cm.

7 cm

4 cm

3 cm

1 cm

Correct Answer: A

Solution:

The length of

PB

is the sum of

PA

and

AB

. Therefore,

PB = PA + AB = 3 + 4 = 7\, \text{cm}

6 cm

8 cm

9 cm

12 cm

Correct Answer: B

Solution:

Let

M

be the midpoint of the chord

AB

. Since

AB

is perpendicular to

OA

, triangle

OAM

is a right triangle with

OM

as the hypotenuse. The length of

AM

is half of

AB

, which is

12

cm. Using the Pythagorean theorem,

OM = \sqrt{OA^2 + AM^2} = \sqrt{10^2 + 12^2} = \sqrt{100 + 144} = \sqrt{244} = 8

cm.

0 degrees

45 degrees

90 degrees

180 degrees

Correct Answer: C

Solution:

The tangent to a circle is perpendicular to the radius at the point of contact, making the angle 90 degrees.

TP is longer than TQ

TP is shorter than TQ

TP is equal to TQ

TP and TQ are unrelated

Correct Answer: C

Solution:

The lengths of the two tangents drawn from an external point to a circle are equal.

12 cm

13 cm

5 cm

8 cm

Correct Answer: A

Solution:

Using the Pythagorean theorem in the right triangle

OTP

, where

OP

is the radius and

OT

is the hypotenuse, we have

TP = \sqrt{OT^2 - OP^2} = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12

cm.

Secant

Chord

Tangent

Diameter

Correct Answer: C

Solution:

A tangent is a line that intersects a circle at exactly one point.

3, \text{cm}

4, \text{cm}

5, \text{cm}

9, \text{cm}

Correct Answer: A

Solution:

Since

TP

is a tangent to the circle at

P

OP \perp TP

. By the Pythagorean theorem,

OT^2 = OP^2 + TP^2

. Substituting the given values,

5^2 = 4^2 + TP^2

. Solving for

TP

, we get

TP = \sqrt{5^2 - 4^2} = \sqrt{25 - 16} = \sqrt{9} = 3\, \text{cm}

7, \text{cm}

8, \text{cm}

9, \text{cm}

10, \text{cm}

Correct Answer: C

Solution:

Let

M

be the midpoint of the chord

AB

. Then

AM = MB = \frac{10}{2} = 5\, \text{cm}

. In the right triangle

OMA

OM^2 + AM^2 = OA^2

. Substituting the given values,

6^2 + 5^2 = OA^2

. Solving for

OA

, we get

OA = \sqrt{6^2 + 5^2} = \sqrt{36 + 25} = \sqrt{61} \approx 7.81\, \text{cm}

The tangents are unequal in length.

The tangents are equal in length.

One tangent is twice the length of the other.

The tangents are perpendicular to each other.

Correct Answer: B

Solution:

The lengths of the two tangents drawn from an external point to a circle are equal.

True or False

Correct Answer: False

Solution:

A circle can be inscribed in a quadrilateral only if the quadrilateral is tangential, meaning the sum of the lengths of opposite sides are equal.

Correct Answer: True

Solution:

An inscribed circle is tangent to each side of the quadrilateral, touching all four sides.

Correct Answer: True

Solution:

From an external point, two tangents can be drawn to a circle, and they are equal in length.

Correct Answer: False

Solution:

A line that intersects a circle at exactly one point is called a tangent, not a secant.

Correct Answer: True

Solution:

The lengths of the tangents from an external point to a circle are always equal.

Correct Answer: True

Solution:

The tangent to a circle is perpendicular to the radius through the point of contact.

Correct Answer: True

Solution:

An inscribed circle is tangent to each side of the triangle, touching each side at exactly one point.

Correct Answer: False

Solution:

There can only be one tangent at any given point on a circle.

Correct Answer: True

Solution:

The lengths of the two tangents from an external point to a circle are equal.

Correct Answer: True

Solution:

By definition, a secant is a line that intersects a circle at two distinct points.

Correct Answer: True

Solution:

A non-intersecting line does not touch the circle at any point, hence it has no common points with the circle.

Correct Answer: True

Solution:

The radius drawn to the point of contact of a tangent is always perpendicular to the tangent.

Correct Answer: True

Solution:

The tangent to a circle is always perpendicular to the radius at the point of contact.

Correct Answer: False

Solution:

A line that does not intersect a circle at any point is called a non-intersecting line, not a tangent.

Correct Answer: False

Solution:

The lengths of the two tangents drawn from an external point to a circle are equal.

Correct Answer: False

Solution:

A line that intersects a circle at one point is called a tangent, not a secant.

Correct Answer: True

Solution:

By definition, a circle is the collection of all points in a plane that are equidistant from a fixed point, known as the center.

Correct Answer: False

Solution:

A tangent to a circle intersects the circle at exactly one point, not two.

Correct Answer: True

Solution:

A secant is defined as a line that intersects a circle at exactly two points.

Correct Answer: False

Solution:

A chord is not necessarily longer than the radius. It can be shorter, equal, or longer depending on its position relative to the center.

Correct Answer: False

Solution:

A line that does not intersect a circle at any point is called a non-intersecting line, not a tangent.

Correct Answer: True

Solution:

At any given point on a circle, there can be only one tangent line.

Correct Answer: False

Solution:

There is only one tangent at a point of the circle.

Correct Answer: True

Solution:

According to the excerpt, a line that does not have any common points with a circle is termed a non-intersecting line.

Correct Answer: False

Solution:

A line that intersects a circle at exactly two points is called a secant, not a tangent.

Correct Answer: False

Solution:

A tangent to a circle intersects the circle at exactly one point.

Correct Answer: False

Solution:

A tangent to a circle touches the circle at exactly one point, not two.

Correct Answer: False

Solution:

A secant line intersects a circle at exactly two points.

Correct Answer: True

Solution:

By definition, the tangent to a circle is perpendicular to the radius at the point where the tangent touches the circle.

Correct Answer: True

Solution:

The tangent from an external point to a circle is always longer than the radius, as it extends from outside the circle to the point of tangency.

Correct Answer: False

Solution:

A line that does not intersect a circle is called a non-intersecting line, not a secant. A secant intersects the circle at two points.

Circles

AI Learning Assistant

Summary

Chapter Summary: Circles

Key Points

Important Properties

Summary of Activities

Theorems

Exercises

Learning Objectives

Detailed Notes

Chapter 10: Circles

10.1 Introduction

10.2 Tangent to a Circle

10.3 Number of Tangents from a Point on a Circle

10.4 Summary

Important Diagrams

Fig. 10.1: Circle and Line Interactions

Fig. 10.10: Tangents and Chords

Fig. 10.12: Circle Inscribed in a Quadrilateral

Fig. 10.14: Triangle with Inscribed Circle

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips

Common Pitfalls

Tips for Success

Practice & Assessment

Multiple Choice Questions

Q1.What is the relationship between a tangent to a circle and the radius drawn to the point of tangency?

Correct Answer: B

Q2.Which of the following statements is true regarding the lengths of two tangents drawn from an external point to a circle?

Correct Answer: B

Q3.In a circle, a chord ABABAB is 12 cm12\, \text{cm}12cm long and is at a distance of 5 cm5\, \text{cm}5cm from the center OOO. What is the radius of the circle?

Correct Answer: C

Q4.In a circle, a chord ABABAB is 14 cm14\, \text{cm}14cm long and is at a distance of 6 cm6\, \text{cm}6cm from the center OOO. What is the radius of the circle?

Correct Answer: B

Q5.What is the term for a line that intersects a circle at exactly one point?

Correct Answer: B

Q6.In a circle, what is the term for a line that does not intersect the circle at any point?

Correct Answer: C

Q7.Which of the following statements is true about a tangent to a circle?

Correct Answer: C

Q8.If a line segment has both endpoints on a circle, what is it called?

Correct Answer: C

Q9.In the context of circles, what is the relationship between the lengths of two tangents drawn from an external point?

Correct Answer: A

Q10.In a circle, if the length from the center to a point on the circle is 5 cm, what is this length called?

Correct Answer: C

Q11.In a circle, which term describes a line segment with both endpoints on the circle?

Correct Answer: C

Q12.A line PQPQPQ is drawn such that it is tangent to a circle at point PPP. If the radius OPOPOP is 7 cm and the distance from the center OOO to an external point TTT is 25 cm, what is the length of the tangent TPTPTP?

Correct Answer: A

Q13.A circle is inscribed in a quadrilateral ABCDABCDABCD such that it touches ABABAB, BCBCBC, CDCDCD, and DADADA at points PPP, QQQ, RRR, and SSS respectively. If AB=8 cmAB = 8\, \text{cm}AB=8cm, BC=10 cmBC = 10\, \text{cm}BC=10cm, CD=12 cmCD = 12\, \text{cm}CD=12cm, find the length of DADADA.

Correct Answer: A

Q14.In a circle, what is the relationship between a tangent and the radius at the point of contact?

Correct Answer: B

Q15.In a circle, if a radius is drawn to the point of tangency, what is the angle between the radius and the tangent?

Correct Answer: C

Q16.A circle is inscribed in a triangle △ABC\triangle ABC△ABC. The points of tangency divide BCBCBC into segments BD=7 cmBD = 7\, \text{cm}BD=7cm and DC=5 cmDC = 5\, \text{cm}DC=5cm. What is the length of BCBCBC?

Correct Answer: B

Q17.If a line is a secant of a circle, how many points does it intersect the circle at?

Correct Answer: C

Q18.If two tangents are drawn from an external point to a circle, what can be said about their lengths?

Correct Answer: B

Q19.What is a secant of a circle?

Correct Answer: B

Q20.If a line intersects a circle at exactly two points, what is this line called?

Correct Answer: B

Q21.In a circle, a tangent at point PPP is perpendicular to the radius OPOPOP. If OP=5 cmOP = 5 \, \text{cm}OP=5cm, what is the length of the tangent from point PPP to a point TTT outside the circle, given that OT=13 cmOT = 13 \, \text{cm}OT=13cm?

Correct Answer: A

Q22.What is the definition of a circle?

Correct Answer: A

Q23.In a circle, a chord PQPQPQ is 16 cm long and is perpendicular to the radius OPOPOP at PPP. If OP=8 cmOP = 8 \, \text{cm}OP=8cm, what is the distance from the center OOO to the midpoint of PQPQPQ?

Correct Answer: A

Q24.Which of the following is NOT a property of a tangent to a circle?

Correct Answer: C

Q25.In a circle, if a line intersects the circle at two distinct points, what is this line called?

Correct Answer: C

Q26.Consider a circle with center OOO and a secant PQPQPQ intersecting the circle at points AAA and BBB. If OA=5 cmOA = 5\, \text{cm}OA=5cm, OB=5 cmOB = 5\, \text{cm}OB=5cm, and the length of the chord AB=8 cmAB = 8\, \text{cm}AB=8cm, what is the distance from OOO to the midpoint of ABABAB?

Correct Answer: B

Q27.If a line PQ intersects a circle at two distinct points, what is this line called?