NCERT Solutions Class 9 Maths Chapter 10 Circles Exercise 10.1

The Anatomy of a Circle and Its Regions

A circle is defined as the collection of all points in a plane that are at a fixed distance from a fixed point in the plane. The fixed point is called the center, and the fixed distance is called the radius. One of the most important concepts in Exercise 10.1 is understanding how a circle interacts with its plane. A circle divides the plane on which it lies into three distinct regions: the interior of the circle (inside), the circle itself (the boundary), and the exterior of the circle (outside). Together, the circle and its interior make up the circular region. If you take any two points on a circle and join them with a straight line segment, this segment is called a chord. When a chord passes directly through the center of the circle, it is called the diameter. The diameter is the longest chord of a circle and is exactly twice the length of the radius ($d = 2r$).

Essential Circle Vocabulary

Arc

A continuous piece of a circle. If the piece is less than a semi-circle, it is a minor arc; if it is greater, it is a major arc.

Segment

The region between a chord and either of its corresponding arcs. A chord divides a circle into a major segment and a minor segment.

Sector

The region between an arc and the two radii joining the center to the endpoints of the arc. It is divided into a major sector and a minor sector.

Semi-circle

When the arc of a circle represents half of the circle's boundary (i.e., when its endpoints are the ends of a diameter).

4-Step Process to Solve Exercise 10.1 Conceptual Questions

1. Identify Key Geometric Terms — Read the problem statement carefully and pinpoint terms like 'chord', 'arc', 'segment', or 'plane'.
2. Recall Mathematical Definitions — Retrieve the exact mathematical definition of the identified term (e.g., remember that a sector requires two radii and an arc, while a segment requires a chord and an arc).
3. Visualize on a Sketchpad — Create a quick visual drawing. Sketch a circle, draw the chord or radii described, and identify the resulting regions.
4. Formulate a Logical Justification — Write down your final choice (True/False or Fill-in-the-blank) and back it up with a clear, definition-based geometric explanation.

CBSE Board Tips: Avoiding Common Terminology Pitfalls

Many Class 9 students lose marks by confusing a Sector with a Segment. Use this easy mental shortcut:

• Se-C-tor is bounded by two Radii and an arc (it looks like a slice of pizza starting from the Center).
• Se-G-ment is bounded by a Chord and an arc (it looks like a sliced-off cap of a circle).

Additionally, always remember that a circle is a two-dimensional plane figure (not a 3D solid like a sphere), and it does not contain its interior region unless we refer to it specifically as a 'circular region'.

Practice Questions with Solutions

• Q: Fill in the blank: The center of a circle lies in _________ of the circle. (interior/ exterior) A: Step 1: Recall the definition of a circle and its plane regions. A circle divides a plane into three parts: interior, exterior, and the boundary. Step 2: The center is the fixed point from which all boundary points are equidistant. This point lies inside the boundary. Final answer: interior
• Q: Fill in the blank: A point, whose distance from the center of a circle is greater than its radius lies in _________ of the circle. (interior/ exterior) A: Step 1: Let the center of the circle be $O$ and the radius be $r$. Let the point be $P$. Step 2: The distance from the center to the point is $OP$. We are given that $OP > r$. Step 3: If $OP = r$, the point lies on the circle. If $OP < r$, the point lies inside. Since $OP > r$, the point must lie outside the boundary. Final answer: exterior
• Q: Write True or False with reasons: Sector is the region between the chord and its corresponding arc. A: Step 1: Analyze the statement. It claims a sector is formed by a chord and an arc. Step 2: Recall the definition of a sector. A sector is the region enclosed by an arc and two radii connecting the center to the endpoints of that arc. Step 3: Recall the definition of a segment. A segment is the region bounded by a chord and an arc. Step 4: Since the statement describes a segment, not a sector, it is mathematically incorrect. Final answer: False. A sector is the region between an arc and two radii joining the center to the endpoints of the arc, whereas the region between a chord and its corresponding arc is a segment.
• Q: Write True or False with reasons: A circle is a plane figure. A: Step 1: Recall the definition of a plane figure. A plane figure is a flat 2-dimensional shape that lies entirely on a single flat surface (plane). Step 2: A circle is drawn on a flat sheet or plane and consists of points that are coplanar. Step 3: Since all points of a circle lie in a single plane, it is a plane figure. Final answer: True. A circle lies entirely in a single two-dimensional plane, making it a plane figure.

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