Circles, Chords, Tangents and Incircles Practice Questions for CAT: 92+ Solved Questions with Step-by-Step Solutions

Solve 92+ Circles, Chords, Tangents and Incircles practice questions for CAT with answers and detailed solutions. Free sample questions below.

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.

Circles, Chords, Tangents and Incircles: Solved Questions with Step-by-Step Explanations (5 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.

Question 3 · Quantitative Ability MCQ

Two identical circles of radius $6$ cm have their centres $62$ cm apart. What is the area, in square cm, of their overlapping region?

A.
36π - 36
B.
18π - 18
C.
18π - 36
D.
9π - 36

Correct Answer:

Step-by-Step Solution

Key idea: equal-circle overlap pattern. The centre distance decides the sector angle.

Step 1: Radius $r = 6$ , centre distance $d = 62$ . In the triangle formed by the two centres and one intersection point, the sides are $6, 6, 62$ .

Step 2: This is a right isosceles triangle, so the angle at each centre between the two intersection radii is $9 0^{\circ}$ .

Step 3: One circular segment inside the overlap is

$sector_{9 0^{\circ}} - triangle_{9 0^{\circ}} .$

Step 4: Sector area:

$\frac{90}{360} π (6)^{2} = 9 π .$

Triangle area:

$\frac{1}{2} (6) (6) = 18.$

One segment $= 9 π - 18$ .

Step 5: The overlap has two such segments:

$2 (9 π - 18) = 18 π - 36.$

Answer: Option C.

Trap: the overlap is not just one sector. It is two segments, each sector minus triangle.

Question 4 · Quantitative Ability MCQ

A circle has radius $6$ cm. One of its chords has length $6$ cm. What is the area, in square cm, of the smaller circular region cut off by this chord?

A.
6π + 9√3
B.
3π - 9√3
C.
6π - 9√3
D.
12π - 9√3

Correct Answer:

Step-by-Step Solution

Key idea: this is a minor segment area question, recognisable because a chord cuts off the smaller region of a circle.

Step 1: Find the central angle. For chord length $c$ , radius $r$ , and central angle $θ$ ,

$c = 2 r sin (\frac{θ}{2}) .$

Here $6 = 2 (6) sin (θ /2)$ , so $sin (θ /2) = \frac{1}{2}$ . For the minor chord, $θ /2 = 3 0^{\circ}$ , hence $θ = 6 0^{\circ}$ .

Step 2: Find the sector area.

$Sector area = \frac{60}{360} π (6)^{2} = 6 π .$

Step 3: Find the triangle area formed by the two radii and the chord.

$Triangle area = \frac{1}{2} (6)^{2} sin 6 0^{\circ} = 18 \cdot \frac{3}{2} = 93 .$

Step 4: The smaller segment is sector minus triangle.

$Segment area = 6 π - 93 .$

Answer: Option C.

Trap check: adding the triangle gives a larger region, not the smaller segment. Assuming $9 0^{\circ}$ ignores the chord-equals-radius condition.

Question 5 · Quantitative Ability MCQ

From an external point $P$ , two tangents $P Q$ and $P R$ are drawn to a circle with centre $O$ , touching it at $Q$ and $R$ . A third tangent cuts $P Q$ at $A$ and $P R$ at $B$ . If $∠ A O B = 6 5^{\circ}$ , what is $∠ A P B$ ?

A.
65°
B.
115°
C.
130°
D.
50°

Correct Answer:

Step-by-Step Solution

Key idea: two tangents from $P$ plus a third tangent create angle-bisector relations at the centre.

Step 1: Let the third tangent touch at $C$ . From $A$ , tangents touch at $Q$ and $C$ , so $O A$ bisects $∠ QO C$ .

Step 2: From $B$ , $O B$ bisects $∠ C O R$ .

Step 3: Hence

$∠ A O B = \frac{1}{2} ∠ QO C + \frac{1}{2} ∠ C O R = \frac{1}{2} ∠ QO R .$

Step 4: Given $∠ A O B = 6 5^{\circ}$ ,

$∠ QO R = 13 0^{\circ} .$

Step 5: For two tangents, angle between tangents and central contact angle are supplementary:

$∠ QP R = 18 0^{\circ} - 13 0^{\circ} = 5 0^{\circ} .$

Since $A$ and $B$ lie on $P Q$ and $P R$ , $∠ A P B = 5 0^{\circ}$ .

Answer: Option D.

Trap: $∠ A O B$ is half of $∠ QO R$ , not directly $∠ A P B$ .

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?

Question 3

Two identical circles of radius 6 cm have their centres 62 cm apart. What is the area, in square cm, of their overlapping region?

Question 4

A circle has radius 6 cm. One of its chords has length 6 cm. What is the area, in square cm, of the smaller circular region cut off by this chord?

Question 5

From an external point P, two tangents P Q and P R are drawn to a circle with centre O, touching it at Q and R . A third tangent cuts P Q at A and P R at B . If ∠ A O B = 6 5^{\circ}, what is ∠ A P B ?

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Circles, Chords, Tangents and Incircles Practice Questions for CAT: 92+ Solved Questions with Step-by-Step Solutions

Chapter Roadmap: Circles, Chords, Tangents and Incircles

Circles, Chords, Tangents and Incircles

🧩 t1 — Chords, Arcs and Circular Segments

📐 t2 — Tangents and Circle Contact Geometry

⭕ t3 — Cyclic Figures, Incircles and Circumcircles

Topic Hero: Chords, Arcs and Circular Segments

Chords, Arcs and Circular Segments

Circles, Chords, Tangents and Incircles: Solved Questions with Step-by-Step Explanations (5 Problems)

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

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