Circles
Circles are the backbone of CAT Geometry. Almost every hard geometry question in CAT, XAT and SNAP leans on a small bundle of circle facts — a chord, a tangent, an inscribed angle, a cyclic quadrilateral — and the candidates who clear the 95+ percentile are simply the ones who recognise which fact applies in two seconds. The beauty of this chapter is that you do not need many theorems; you need to wield about nine of them with total confidence. The perpendicular from the centre bisects a chord, equal chords sit equidistant from the centre, a tangent meets the radius at 90°, the two tangents from an external point are equal, the power of a point ties intersecting chords and secants together, the alternate segment theorem links a tangent to the inscribed angle, opposite angles of a cyclic quadrilateral are supplementary, and the angle at the centre is twice the angle at the circumference. This chapter walks through each with CAT-style worked problems, the fastest recognition cues, and the traps that quietly drain marks. Master these and circle questions flip from intimidating to free.
Topics
⚡ CAT shortcuts & speed methods
The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.
- For any chord problem, drop the perpendicular from the centre and use chord = 2√(r² − d²) — it converts the figure into one right triangle.
- Tangent length from an external point P is √(OP² − r²); the radius-to-tangent 90° is your built-in right angle.
- Power of a point unifies three cases: chords PA×PB = PC×PD, secants PA×PB = PC×PD, tangent–secant PT² = PA×PB.
- See a diameter? Mark 90° at any third point instantly (angle in a semicircle).
- Angle at centre = 2 × angle at circumference; angles in the same segment are equal — use them to migrate a known angle across the circle.
- Tangent meets a chord? Apply the alternate segment theorem at once: that angle equals the inscribed angle across the circle.
⚠️ Common mistakes & traps
CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Using the full chord instead of the half-chord in chord = 2√(r² − d²) — d, the half-chord, and r form the right triangle, not the full chord.
- Adding chord distances when the chords are on opposite sides but subtracting requires the same side (and vice versa).
- Writing PT = √(OP² + r²) instead of √(OP² − r²) — the radius is a leg, OP is the hypotenuse.
- Forgetting that opposite (not adjacent) angles of a cyclic quadrilateral are supplementary.
- Doubling the inscribed angle to get the central angle but then doubling again, or halving the wrong way — central is the larger (2×).
📈 CAT exam insight & PYQ analysis
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
Chapter test
🏆 Vidaara CAT success checklist
You have truly mastered Circles when you can tick every box below.
- Recall every formula in this chapter without looking them up
- Solve each topic’s practice set with at least 80% accuracy
- Use the chapter shortcuts to cut your solving time in half
- Spot and avoid every common trap listed above
- Score 80%+ on the timed chapter test
📋 Chapter mastery scorecard
Track where you stand. Aim for the target before moving to the next chapter.
| Skill checkpoint | Target |
|---|---|
| Concept theory & formulas understood | 100% |
| Topic practice sets attempted (4 topics) | 4/4 |
| Best topic-test score | — → 80%+ |
| Chapter test score | — → 80%+ |
| Flashcards drilled to instant recall | 12 cards |
Formula Reference Sheet
Chords, tangents & power of a point
| Perpendicular from centre bisects chord | OM ⊥ AB ⇒ AM = MB |
|---|---|
| Chord length from distance d to centre | chord = 2√(r² − d²) |
| Tangent length from external point P | PT = √(OP² − r²) |
| Two intersecting chords | PA × PB = PC × PD |
| Secant–secant from external P | PA × PB = PC × PD |
| Tangent–secant from external P | PT² = PA × PB |
Angles in a circle
| Angle at centre vs circumference | ∠centre = 2 × ∠circumference (same arc) |
|---|---|
| Angle in a semicircle | Angle on a diameter = 90° |
| Cyclic quadrilateral opposite angles | ∠A + ∠C = ∠B + ∠D = 180° |
| Alternate segment theorem | angle between tangent & chord = inscribed angle in alternate segment |
| Angles in the same segment | equal (subtend the same arc) |
| Exterior angle of cyclic quad | = interior opposite angle |