Last Updated on September 8, 2024 by XAM CONTENT

Hello students, we are providing case study questions for class 9 maths. Case study questions are the new question format that is introduced in CBSE board. The resources for case study questions are very less. So, to help students we have created chapterwise case study questions for class 9 maths. In this article, you will find case study questions for CBSE Class 9 Maths Chapter 9 Circles. It is a part of Case Study Questions for CBSE Class 9 Maths Series.

Chapter | Circles |

Type of Questions | Case Study Questions |

Nature of Questions | Competency Based Questions |

Board | CBSE |

Class | 9 |

Subject | Maths |

Useful for | Class 9 Studying Students |

Answers provided | Yes |

Difficulty level | Mentioned |

Important Link | Class 9 Maths Chapterwise Case Study |

### Case Study Questions on Circles

**Questions**

Government of India is working regularly for the growth of handicapped persons. For these three STD booths situated at point P, Q and R are as shown in the figure, which are operated by handicapped persons. These three booths are equidistant from each other as shown in the figure.

On the basis of the above information, solve the following questions:

Q. 1. Which type of Î”PQR in the given figure?

Q. 2. Measure angle âˆ QOR.

Q. 3. Find the value of âˆ OQR.

Q. 4. Is it true that points P, Q and R lie on the circle?

**Solutions:**

1. Given $P, Q$ and $R$ are equidistant. It means their distances are equal.

Note:

In an equilateral triangle, length of all three sides are equal.

So, $\triangle P Q R$ is an equilateral triangle.

2. Since, $\triangle P Q R$ is an equilateral triangle.

$$

\therefore \angle \mathrm{PQR}=\angle \mathrm{PRQ}=\angle \mathrm{QPR}=60^{\circ}

$$

The angle subtended by an arc at the centre is double the angle subtended by it any point on the remaining part of the circle.

$$

\therefore \angle Q O R=2 \angle Q P R=2 \times 60^{\circ}=120^{\circ}

$$

3. In $\triangle O Q R$,

$$

O Q=O R[\text { Radii of a circle })

$$

$\Rightarrow \angle \mathrm{ORQ}=\angle \mathrm{OQR}$ [Angles opposite to equal sides of a triangle are equal]

Using angle sum property of a triangle,

$$

\begin{aligned}

& \angle \mathrm{OQR}+\angle \mathrm{ORQ}+\angle \mathrm{QOR}=180^{\circ} \\

& \Rightarrow \angle \mathrm{OQR}+\angle \mathrm{OQR}+120^{\circ}=180^{\circ}

\end{aligned}

$$

$$

\begin{gathered}

\Rightarrow 2 \angle O Q R=60^{\circ} \\

\therefore \angle O Q R=30^{\circ}

\end{gathered}

$$

4. Yes, it is true that points $P, Q$ and $R$ lie on the circle.

### Understanding Circles

**Circle: **A collection of all points in a plane which are at a constant distance from a fixed point. The fixed point is called the centre of the circle and the constant distance is called the radius.

In figure, O is the centre, OA is the radius and AB is the diameter of the circle.

**Chord: **A line segment joining any two points on the circle. In figure, AC is the chord.

**Diameter: **Longest chord of the circle that passes through the centre of the circle.

**Circumference: **Length of the boundary of a circle.

**Arc: **Any part of the circumference of a circle.

In figure, PRQ is minor arc represented as PRQ and PSQ is major arc represented as PSQ.

**Semi-circle: **Parts of a circle that are divided by its diameter.

**Segment: **The region between a chord and either of its arcs (major or minor). The segment formed with a minor arc is called minor segment and that formed with a major arc is called major segment.

**Sector: **The region enclosed by an arc and the two radii joining the centre to the end points of the arc.

The sector corresponding to minor arc is called minor sector i.e., AOB and that corresponding to major arc is called major sector. i.e., AOBCA

### Also check

- Heron’s Formula Class 9 Case Study Questions Maths Chapter 10
- Circles Class 9 Case Study Questions Maths Chapter 9
- Quadrilaterals Class 9 Case Study Questions Maths Chapter 8
- Triangles Class 9 Case Study Questions Maths Chapter 7
- Lines and Angles Class 9 Case Study Questions Maths Chapter 6
- Introduction to Euclidâ€™s Geometry Class 9 Case Study Questions Maths Chapter 5
- Linear Equations in Two Variables Class 9 Case Study Questions Maths Chapter 4
- Coordinate Geometry Class 9 Case Study Questions Maths Chapter 3
- PolynomialsÂ Class 9 Case Study Questions Maths Chapter 2
- Number SystemsÂ Class 9 Case Study Questions Maths Chapter 1

### Topics from which case study questions may be asked

- Equal chords of a circle subtend equal angles at the center and (motivate) its converse.
- The perpendicular from the center of a circle to a chord bisects the chord and onversely, the line drawn through the center of a circle to bisect a chord is perpendicular to the chord.
- Equal chords of a circle (or of congruent circles) are equidistant from the center (or their respective centers) and conversely.
- The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.
- Angles in the same segment of a circle are equal.
- If a line segment joining two points subtends equal angle at two other points lying on the same side of the line containing the segment, the four points lie on a circle.
- The sum of either of the pair of the opposite angles of a cyclic quadrilateral is 180Â° and its converse.

Circles having same centre are called concentric circles.

Case study questions from the above given topic may be asked.

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### Frequently Asked Questions (FAQs) on Circles Case Study

#### Q1: What is the definition of a circle in Class 9 Maths?

A1: A circle is defined as the collection of all points in a plane that are at a fixed distance (radius) from a fixed point called the center. In Class 9, the concept of a circle is used to study properties related to chords, tangents, arcs, and sectors.

#### Q2: What is the difference between a chord and a diameter?

A2: A chord is a line segment joining two points on the circumference of a circle, whereas the diameter is a special chord that passes through the center of the circle. The diameter is the longest chord of the circle and is equal to twice the radius.

#### Q3: **What are tangents and how are they important in this chapter?**

A3: A tangent to a circle is a line that touches the circle at exactly one point. Tangents are important in this chapter as you learn properties like “a tangent to a circle is perpendicular to the radius at the point of contact.” You also explore problems involving two tangents drawn from an external point.

#### Q4: **What are the main properties of circles discussed in Class 9 Maths?**

A4: The main properties include:

(i) The radius is perpendicular to the tangent at the point of contact.

(ii) Equal chords are equidistant from the center.

(iii) The angle subtended by a chord at the center of the circle is twice the angle subtended by the same chord at any other point on the circumference.

#### Q5: What are cyclic quadrilaterals, and how are they related to circles?

A5: A cyclic quadrilateral is a four-sided figure where all its vertices lie on the circumference of a circle. The sum of the opposite angles in a cyclic quadrilateral is always 180Â°. This property is crucial in solving various geometric problems involving circles.

#### Q6: How can we prove that the angle subtended by a diameter on the circumference is 90 degrees?

A6: This is a key result derived from the properties of circles. Using the theorem “the angle subtended by a chord at the center is twice the angle subtended at any other point on the circumference,” and since a diameter subtends a straight angle (180Â°) at the center, it subtends a right angle (90Â°) on the circumference.

#### Q7: How many tangents can be drawn from a point outside a circle?

A7: From any point outside a circle, exactly two tangents can be drawn. These tangents have equal lengths from the external point to their points of contact on the circle.

#### Q8: What are the most important theorems in Chapter 9 of Class 9 Maths?

A8: The two key theorems in this chapter are:

(i) The perpendicular from the center of a circle to a chord bisects the chord.

(ii) The length of tangents drawn from an external point to a circle are equal.

#### Q9: **Are there any online resources or tools available for practicing Circles case study questions?**

A9: We provide case study questions for CBSE Class 9 Maths on our website. Students can visit the website and practice sufficient case study questions and prepare for their exams. If you need more case study questions, then you can visit Physics Gurukul website. they are having a large collection of case study questions for all classes.