Complete solutions with diagrams, explanations & exam tips for every question
CBSETelangana BoardAndhra Pradesh Board2 Questions · 15 Parts
About Exercise 12.1
Exercise 12.1 is the first exercise in Chapter 12 – Circles of the Class 9 Mathematics textbook, prescribed by the Telangana, Andhra Pradesh, and CBSE boards. It tests whether students can identify and name the different parts of a circle from a diagram, and whether they understand the precise definitions well enough to judge statements as true or false.
This exercise has two questions: Question 1 asks you to name eight parts of a labelled circle diagram, and Question 2 presents seven statements for you to classify as True or False with reasoning. Together these 15 parts cover every key vocabulary term introduced in the Introduction to Circles.
Q1 — Identify Parts (8 parts)Q2 — True or False (7 statements)Marks: 1–2 per part in board exams
Question 1 — Reference Diagram
The question refers to the figure below, where O is the centre of the circle. Points A, B, C, D lie on the circle; AB is a diameter; X marks reference positions; and the region near chord AD is shaded. Study this diagram carefully before reading each answer — every part name makes immediate sense once you can see the figure clearly.
Figure for Question 1 — O is the centre
AB = diameter (red dashed) · AC & AD = chords · Shaded region = minor segment
Question 1 — Solutions: Naming Parts of the Circle
The question asks you to name each of the following from the figure. Here is a detailed answer for each part, explaining why it has that name — not just what the name is.
Part (i)
Line segment AO
Answer: Radius
AO connects point A (on the circle) to O (the centre). By definition, any line segment from the centre to the circumference is a radius. All radii of the same circle are equal in length.
Part (ii)
Line segment AB
Answer: Diameter
AB is a chord that passes through the centre O. Such a chord is called the diameter — the longest possible chord. In the figure, AB is shown as a dashed line going from A through O to B.
Part (iii)
Arc BC (curved part)
Answer: Minor Arc
The arc BC is the shorter curved path between B and C along the circle's boundary. Because it is the smaller of the two arcs connecting B and C, it is called the minor arc BC.
Part (iv)
Line segment AC
Answer: Chord
AC joins two points A and C both lying on the circle. A line segment joining any two points on a circle is called a chord. Since AC does not pass through the centre O, it is an ordinary chord (not a diameter).
Part (v)
Arc DCB (curved part)
Answer: Major Arc
The arc DCB travels the longer route from D through C to B along the circumference. Since it is the larger of the two arcs connecting B and D (via C), it is the major arc DCB.
Part (vi)
Arc ACB (curved part)
Answer: Semicircle
Since AB is a diameter, it divides the circle into two exactly equal halves. Arc ACB is one of those equal halves. Any arc that is exactly half of the full circle (endpoints being ends of a diameter) is called a semicircle.
Part (vii)
Line segment AD
Answer: Chord
AD connects two points A and D both on the circumference. Like AC, this is a chord. It does not pass through O, so it is not a diameter. The shaded region in the figure is bounded partly by this chord AD.
Part (viii)
Shaded region
Answer: Minor Segment
The shaded (hatched) region lies between chord AD and the minor arc AD. This enclosed region — bounded by a chord and the smaller arc — is called a minor segment. If the arc were the major arc instead, it would be a major segment.
Part
Given
Answer
Reason in Brief
(i)
AO
Radius
Centre O to point A on circle
(ii)
AB
Diameter
Chord through centre O; longest chord
(iii)
arc BC
Minor Arc
Shorter arc between B and C
(iv)
AC
Chord
Joins two points on circle; not through O
(v)
arc DCB
Major Arc
Longer arc from D via C to B
(vi)
arc ACB
Semicircle
Half circle; AB is a diameter
(vii)
AD
Chord
Joins A and D on the circle; not through O
(viii)
Shaded region
Minor Segment
Region between chord AD and minor arc AD
💡 Pattern to notice: Parts (i)–(ii) involve line segments. Parts (iii), (v), (vi) involve arcs (curved boundaries). Parts (iv) and (vii) are chords. Part (viii) is a region. Grouping them this way helps you answer similar questions quickly in exams.
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Question 2 — True or False: Complete Solutions with Reasoning
This question tests whether you understand the definitions, not just the names. For each statement, you must decide if it is True or False — and in board exams, you often need to give a one-line reason. Here is a full explanation for all seven statements.
(i)
A circle divides the plane on which it lies into three parts.
The three parts are: (1) the interior — all points inside the circle, (2) the circumference — all points exactly on the boundary, and (3) the exterior — all points outside the circle. This is a fundamental property of circles and always holds true.
✓ True
(ii)
The region enclosed by a chord and the minor arc is the minor segment.
By definition, when a chord divides a circle, the region between the chord and the shorter (minor) arc is called the minor segment. This is exactly what the statement says, so it is True.
✓ True
(iii)
The region enclosed by a chord and the major arc is the major segment.
This mirrors (ii). The region between a chord and the longer (major) arc is called the major segment. The statement correctly describes this definition, so it is True.
✓ True
(iv)
A diameter divides the circle into two unequal parts.
This is False. The diameter is the longest chord and it always passes through the centre. Because it passes through the centre, it divides the circle into two perfectly equal halves, each called a semicircle. Two unequal parts would result only from a chord that does NOT pass through the centre.
✗ False
(v)
A sector is the area enclosed by two radii and a chord.
This is False. A sector is bounded by two radii and an arc (not a chord). The region bounded by two radii and a chord would actually be a triangle (not a recognised circle part). A segment — not a sector — involves a chord. Remember: sector = two radii + arc; segment = chord + arc.
✗ False
(vi)
The longest of all chords of a circle is called a diameter.
This is True. Among all chords of a circle, the diameter is always the longest because it is the only chord that passes through the centre. Any other chord is shorter because it cannot span the full width of the circle.
✓ True
(vii)
The midpoint of any diameter of a circle is the centre.
This is True. A diameter passes through the centre, with one endpoint on each side of the circle. Since the radius from centre to each endpoint is equal, the centre is exactly halfway along the diameter — making it the midpoint of the diameter.
✓ True
Statement
Verdict
Key Reason
(i) Circle divides plane into 3 parts
True
Interior + Circumference + Exterior
(ii) Chord + minor arc = minor segment
True
Correct definition of minor segment
(iii) Chord + major arc = major segment
True
Correct definition of major segment
(iv) Diameter divides circle into 2 unequal parts
False
Diameter → two equal semicircles
(v) Sector = 2 radii + chord
False
Sector = 2 radii + arc (not chord)
(vi) Longest chord = diameter
True
Diameter spans full width through centre
(vii) Midpoint of diameter = centre
True
Centre is equidistant from both ends
The Most Important Distinction — Sector vs Segment
Statement (v) in Question 2 trips up many students because "sector" and "segment" sound similar. Here is a clear visual comparison to lock in the difference permanently.
🍕 Sector
Bounded by 2 radii + arc Includes the centre O
🍰 Segment
Bounded by chord + arc Does NOT include centre
Sector = 2 radii + arc (pointed tip at centre O) Segment = 1 chord + arc (flat edge — no centre)
⛔ Most Common Exam Error: Writing "sector = two radii and a chord" (as Statement v says) is WRONG. The correct definition is sector = two radii and an arc. If you mix up arc and chord here, you lose 1–2 marks. Always remember: sector has a pointed tip at the centre; segment has a flat chord edge.
Common Mistakes to Avoid in Exercise 12.1
Confusing radius and chord (Q1-i vs Q1-iv): A radius always starts at the centre O. A chord connects two points on the circle's boundary — it may or may not pass through O. If it passes through O, it becomes a diameter.
Calling arc ACB a "semicircle" without justification: The key reason arc ACB is a semicircle is that its two endpoints A and B are the ends of a diameter. Without this condition, any arc could be confused for a semicircle just because it "looks like half."
Incorrectly identifying the shaded region (Q1-viii): The shaded region is a segment (chord + arc), not a sector (radii + arc). Make sure to check whether the boundary includes the centre point O or not.
Statement (iv) error — thinking diameter creates unequal parts: A non-diameter chord creates unequal parts; a diameter always creates two equal semicircles. This is a favourite true/false trap in Telangana and AP board exams.
Statement (v) error — sector vs segment confusion: The boundary of a sector has two straight sides (radii) meeting at a point (the centre) plus a curved side (arc). The boundary of a segment has one straight side (chord) plus one curved side (arc) — no centre involved.
What Exercise 12.1 Prepares You For
The identification and definition skills from Exercise 12.1 are the foundation for all the theorems in the rest of Chapter 12. When you study results like "equal chords are equidistant from the centre" or "the angle subtended by a chord at the centre is twice the angle at any other point on the circle," you will constantly need the precise meanings of chord, arc, radius, and segment established here.
The true/false reasoning format from Question 2 also directly mirrors the 1-mark and 2-mark questions that appear in both Telangana SSC and AP SSC board examinations. Practising the justifications — not just the True/False verdict — is essential for scoring full marks. For further practice on circle properties, see the Exercise 12.2 — Chord Properties page, which builds directly on the vocabulary established here. You can also revisit the Introduction to Circles if any definition feels unclear.
📐 Board Exam Tip (CBSE, Telangana & AP): In Q1-type questions, always name the element precisely — don't just write "arc"; specify whether it is a minor arc, major arc, or semicircle. For Q2-type true/false, always write the reason in one clear sentence — the verdict alone is rarely awarded full marks.
