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

Circles, Chords, Tangents and Incircles short notes for CAT: 47 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

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.

Chord Length from Radius and Distance

Chord length formula

Radius $r$ , distance of chord from centre $d$ .

Half chord

\frac{chord}{2} = r^{2} - d^{2}

Full chord

chord = 2 r^{2} - d^{2}

Exam translation: distance from centre increases

\Rightarrow

chord becomes shorter.

Equal Chords, Equal Distances

Same circle comparison rule

Equal chords	are equally distant from the centre.
Equal distances	produce equal chords.
Larger distance	produces a smaller chord.

Use this mainly when two chords are parallel or when their distances from the centre are compared.

Chord Length from Central Angle

Central angle gives chord

If chord $A B$ subtends angle $θ$ at centre $O$ :

A B = 2 r sin \frac{θ}{2}

Most useful for angles like $6 0^{\circ}$ , $9 0^{\circ}$ , $12 0^{\circ}$ .

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 short notes in this unit

chapter

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

Circles, Chords, Tangents and Incircles short notes for CAT: 47 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questi

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?

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