Circles, Chords, Tangents and Incircles Notes for CAT: Concepts, Formulas, Worked Examples & Practice

Circles, Chords, Tangents and Incircles notes for CAT: 54 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Circles, Chords, Tangents and Incircles

CAT QA Geometry

Circles, Chords, Tangents and Incircles

A 3-step journey from measuring a circle to solving full exam geometry.

chapter PYQs

①

🧩 t1 — Chords, Arcs and Circular Segments

You learn how chord length, distance from centre, central angle, sector area, and segment area talk to each other.

2 own-course PYQs Importance: moderate Master: circle measurement

②

📐 t2 — Tangents and Circle Contact Geometry

You move from inside chords to outside touching lines and contact-based angle geometry.

2 own-course PYQs Importance: moderate

③

⭕ t3 — Cyclic Figures, Incircles and Circumcircles

You combine circles with triangles, rectangles, quadrilaterals, incircles, and circumcircles.

6 own-course PYQs Highest chapter weight

End goal: by the end of this chapter, you should see a circle question and quickly decide whether it is about measurement inside the circle, touch/contact outside the circle, or circle mixed with polygons.

Topic Hero: Chords, Arcs and Circular Segments

Selected Topic

Chords, Arcs and Circular Segments

A chord cuts the circle. The arc bends above it. The segment is the curved slice between them.

CAT skill: area + angle 2 direct PYQs Moderate frequency

One-line hook: Most chord-and-segment problems become easy once you draw the radius to the chord’s midpoint.

The Three Objects: Chord, Arc, Segment

Chord, Arc, Segment: do not mix them

➖

Chord

Straight line joining two points on the circle.

🌙

Arc

Curved boundary between the two points.

🍕

Segment

Area trapped between the chord and the arc.

If a chord cuts a circle into two regions, the smaller region is usually the minor segment.

The Master Move: Drop a Perpendicular from Centre

Core theorem

Centre perpendicular to chord bisects the chord

For chord $A B$ , if $O M ⊥ A B$ , then $A M = M B$ .

O A^{2} = O M^{2} + A M^{2}

radius squared equals distance from centre squared plus half-chord squared.

Circles, Chords, Tangents and Incircles: Solved Questions with Step-by-Step Explanations (2 Problems)

Question 1 · Quantitative Ability MCQ

Three identical circles of radius $3$ cm touch each other externally. A smaller circle is placed in the central gap so that it touches all three circles externally. What is its radius, in cm?

A.
2√3 - 3
B.
3 - 2√3
C.
2√3 + 3
D.
6 - 2√3

Correct Answer:

Step-by-Step Solution

Key idea: inner circle in the symmetric gap between three equal touching circles.

Step 1: Centres of the three given circles form an equilateral triangle of side

$3 + 3 = 6.$

Step 2: The gap circle centre is the triangle centre. Distance from triangle centre to a vertex is the circumradius:

$\frac{6}{3} = 23 .$

Step 3: If the small radius is $x$ , external touch gives

$3 + x = 23 .$

Step 4: Solve:

$x = 23 - 3.$

Answer: Option A.

Trap: using subtraction here belongs to the outer enclosing circle case. The inner gap circle touches externally, so radii add.

Question 2 · Quantitative Ability MCQ

In a circle of radius $10$ cm, two parallel chords have lengths $12$ cm and $16$ cm. The chords may be placed on either side of the centre. What is the greatest possible area, in square cm, of the trapezium formed by joining their corresponding endpoints?

A.
28
B.
98
C.
280
D.
196

Correct Answer:

Step-by-Step Solution

Key idea: this is a parallel-chords trapezium question with a maximisation layer. The parallel sides are the chords, so the area is

$\frac{1}{2} (sum of chord lengths) \times (distance between chords) .$

Step 1: Find the distance of each chord from the centre. For chord length $c$ and radius $r$ ,

$d = r^{2} - (\frac{c}{2})^{2} .$

For the $12$ cm chord:

$d_{1} = 1 0^{2} - 6^{2} = 64 = 8.$

For the $16$ cm chord:

$d_{2} = 1 0^{2} - 8^{2} = 36 = 6.$

Step 2: Maximise the distance between the chords. If they are on the same side, separation is $8 - 6 = 2$ . If they are on opposite sides, separation is $8 + 6 = 14$ . The greatest possible height is therefore $14$ .

Step 3: Compute the maximum area.

$Area = \frac{1}{2} (12 + 16) (14) = 14 \times 14 = 196.$

Answer: Option D.

Trap check: using the same-side separation gives the minimum area, while using the diameter as height ignores the actual chord positions.

More notes in this unit

chapter

Circles, Chords, Tangents and Incircles Notes for CAT: Concepts, Formulas, Worked Examples & Practice

Circles, Chords, Tangents and Incircles notes for CAT: 54 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

A question from this chapter

Question 1

Three identical circles of radius 3 cm touch each other externally. A smaller circle is placed in the central gap so that it touches all three circles externally. What is its radius, in cm?

Question 2

In a circle of radius 10 cm, two parallel chords have lengths 12 cm and 16 cm. The chords may be placed on either side of the centre. What is the greatest possible area, in square cm, of the trapezium formed by joining their corresponding endpoints?

Free preview ends here

Login to view the complete notes

Creating an account is free. You get the rest of this chapter, step-by-step solutions, and a study plan built around the topics you are actually weak at.

Why MastersUp

Personalised first. High quality throughout.

Most platforms hand everyone the same content. Here the content moves with your performance, topic by topic.

Built around you, not around a syllabus PDF

Every answer you give moves your topic-level intelligence rate. The next question, the next revision card and tomorrow's plan all change with it.

Revision that hits your weak spots

We only revise topics you have actually attempted and are still below the safe bar on — never the same chapter on repeat.

Questions calibrated to the real exam

Each question carries a measured toughness. You are served a rung above your current level, so practice keeps stretching you.

Notes written for recall, not for volume

Full lesson cards for first study, curated short-note cards for the last mile — with derivations, traps and exam patterns marked.

One place for everything

Notes, chapter practice, previous-year questions, test series and full-length papers — all feeding one picture of your preparation.

Honest progress

No vanity streaks. Progress here means chapters mastered and accuracy that held up on harder questions.

Unlock the whole course

Full notes and short notes, the complete question bank with worked solutions, mock tests, full-length papers, and an adaptive plan that rebuilds itself as you improve.

View all plans Start free