Circles for CAT 2026: Properties, Tangents + 12 Solved Qs
A recognition-first cheatsheet for CAT 2026 circles covering all 8 core properties, the power-of-a-point cluster (intersecting chords, secant-tangent, two-secant), the alternate segment theorem, the angle-in-semicircle and cyclic quadrilateral families, and 12 solved CAT-level questions across MCQ and TITA. Built to convert circles from a derivation chapter into a trigger-phrase recognition chapter that finishes in 60 to 90 seconds per question.
Circles for CAT 2026: Properties, Tangents + 12 Solved Qs
The aspirants who lose marks on circles CAT 2026 almost never lose them on the algebra. They lose them on recognition. A perpendicular dropped from the centre to a chord is a chord bisector. A tangent at a point of contact is perpendicular to the radius. Two chords intersecting inside the circle satisfy a product rule. The candidate who sees the figure and instantly names the theorem finishes in 60 seconds; the one who tries to derive the relationship from triangle congruence burns 90 seconds and arrives at the same answer. Circles on CAT is a theorem-recognition test wrapped in a geometry shell.
This guide covers circles CAT 2026 as a recognition-first playbook: the 8 core properties every CAT cycle tests, the tangent and chord theorems (including the alternate segment rule), the intersecting chords and power-of-a-point setups, the angle-subtended family (semicircle, central versus inscribed, cyclic quadrilateral), and 12 solved CAT-level questions across MCQ and TITA. It completes the geometry cluster alongside the triangles cheatsheet, the coordinate geometry guide, and the mensuration formulas reference.
Circles contribute 2 to 3 marks per CAT cycle. Eight core properties cover roughly 90 percent of setups. The four recurring triggers: perpendicular from centre bisects a chord; tangent at a point is perpendicular to the radius; tangent lengths from an external point are equal; angle in a semicircle is 90 degrees. Power of a point unifies intersecting chords, secant-tangent, and two-secant setups. Recognition first. Computation second.
The 8 Core Circle Properties for CAT 2026
These eight properties cover roughly 90 percent of CAT circle questions across the last 10 cycles. Memorise the trigger phrase for each one. When the problem mentions a chord, a tangent, an angle on a circle, or two chords crossing, the right property activates in under five seconds.
| # | Property | CAT trigger phrase |
|---|---|---|
| 1 | Perpendicular from centre to a chord bisects the chord | Chord with perpendicular dropped from centre |
| 2 | Equal chords are equidistant from the centre | Two chords with equal length |
| 3 | Tangent at a point is perpendicular to the radius at that point | Tangent meets the circle at a single point |
| 4 | Tangents from an external point are equal in length | Two tangents from a single external point |
| 5 | Angle in a semicircle is 90 degrees | Diameter as one side of an inscribed triangle |
| 6 | Central angle equals twice the inscribed angle on the major arc | One angle at the centre, another on the circumference |
| 7 | Angles in the same segment are equal | Multiple inscribed angles on the same chord |
| 8 | Opposite angles of a cyclic quadrilateral sum to 180 degrees | Four points on a circle forming a quadrilateral |
Build a one-page property-trigger card. Left column: the trigger phrase from the question. Right column: the theorem that activates. The mental routine should finish before pencil touches paper. The 5-second recognition is the difference between a 90-second solve and a 150-second solve.
Tangent and Chord Theorems: The Power-of-a-Point Cluster
Three theorems share a single underlying idea: the power of a point with respect to a circle is constant. CAT tests all three variants, but the candidate who sees them as one family solves each in 30 seconds.
PA × PB = PC × PD
Use when two chords cross and three of the four segment lengths are given.
PT² = PX × PY
Use when one tangent and one secant meet at the same external point.
PA × PB = PC × PD
Use when two secants from the same external point are given.
Angle PTA = inscribed angle subtended by TA in the alternate segment
Use when a tangent and a chord meet at the tangent point and the question asks for an arc angle on the opposite side.
Myth
To find a tangent length from an external point, you need the Pythagorean theorem and the radius.
Reality
If a secant is also given, the secant-tangent rule (PT² = PX × PY) skips Pythagoras and the radius entirely. One line of arithmetic; no coordinates needed.
The Angle-Subtended Family: Semicircle, Central, Inscribed, Cyclic
Properties 5 through 8 form a single family. They all describe how angles relate when the vertex sits on a circle, at the centre, or on a chord. CAT picks one variant per question, but the underlying idea is the same: an arc determines a fixed angle relationship across every point on the circle that sees it.
Angle APB = 90 degrees for every such P.
Trigger: diameter mentioned as a chord; an inscribed triangle has the diameter as one side.
theta = 2 × phi
Inscribed angles on the major arc are half the central angle; inscribed angles on the minor arc are (180 minus phi).
Angle A + Angle C = 180 degrees · Angle B + Angle D = 180 degrees
The exterior angle equals the opposite interior angle. Useful for hidden right triangles inside a quadrilateral.
CAT often gives an inscribed angle on the minor arc and expects the central angle on the major arc, or vice versa. The relationship theta equals 2 phi only holds when both vertices look at the same arc. If the inscribed angle vertex is on the minor arc, the inscribed angle is 180 minus half the central angle. Always confirm which arc the inscribed vertex sits on before doubling or halving.
Want a topic-priority map showing where circles sits among the 30 CAT Quant chapters at your current level?
Map My CAT Quant Topic Priority12 Solved CAT-Level Circle Questions (MCQ + TITA)
Questions span chord bisection, tangent length, intersecting chords, alternate segment, angle in a semicircle, central versus inscribed, and cyclic quadrilateral setups. Target time: 60 to 90 seconds per MCQ; 90 to 120 seconds for the harder TITA setups.
A chord of length 24 cm is at a perpendicular distance of 5 cm from the centre of a circle. Find the radius.
Half-chord = 12. Radius = √(12² + 5²) = √(144 + 25) = 13 cm.
A tangent from an external point P to a circle of radius 6 cm has length 8 cm. Find the distance from P to the centre.
PO² = 6² + 8² = 100. PO = 10 cm.
Two tangents from an external point P touch a circle at A and B. If PA = 9 cm, find PB.
Tangents from a common external point are equal. PB = 9 cm.
Two chords AB and CD of a circle intersect at P inside the circle. PA = 4, PB = 9, PC = 6. Find PD.
PA × PB = PC × PD. 4 × 9 = 6 × PD. PD = 6.
From an external point P, a tangent of length 12 cm touches a circle, and a secant from P cuts the circle at X and Y with PX = 8 cm. Find PY.
PT² = PX × PY. 144 = 8 × PY. PY = 18 cm.
A point P on a circle is joined to the endpoints of a diameter. Find the measure of angle at P.
Angle in a semicircle is a right angle. Angle at P = 90 degrees.
A chord AB subtends an angle of 70 degrees at a point P on the major arc. Find the angle subtended by AB at the centre.
Central angle = 2 × inscribed angle on the major arc = 2 × 70 = 140 degrees.
In a cyclic quadrilateral ABCD, angle A = 65 degrees and angle B = 100 degrees. Find angle C and angle D.
Opposite angles sum to 180 degrees. Angle C = 180 - 65 = 115. Angle D = 180 - 100 = 80. Answer: C = 115, D = 80.
A tangent PT at T meets a chord TA of a circle. The angle between the tangent and the chord is 55 degrees. Find the inscribed angle in the alternate segment.
Alternate segment theorem: inscribed angle in the alternate segment = angle between tangent and chord = 55 degrees.
From external point P, two secants cut the circle at (A, B) and (C, D). PA = 3, PB = 12, PC = 4. Find PD.
PA × PB = PC × PD. 3 × 12 = 4 × PD. PD = 9.
Two chords of a circle are equal in length. The first is at a distance of 7 cm from the centre. Find the distance of the second chord from the centre.
Equal chords are equidistant from the centre. Distance = 7 cm.
A circle of radius 5 cm has a chord of length 6 cm. Find the distance of the chord from the centre.
Half-chord = 3. Distance = √(5² - 3²) = √16 = 4 cm.
Drill Cadence: From Day 1 to CAT Eve
Circles lock in across 10 to 12 hours of focused work spread across two to three weeks. The cadence below maps to the recognition-first approach: trigger-phrase identification gets the bulk of the time; computation drills follow.
Week 2 (4 hrs): 25 questions across chord, tangent, and angle-subtended setups; classify each by trigger phrase; aim for 5-second recognition.
Week 3 (3 hrs): 15 hybrid questions mixing circles with triangles, mensuration, or coordinate geometry.
Mock window (weekly 30-min): Drill 4 mixed circle questions; classify recognition errors versus computation errors. The recognition error column should shrink to zero by week three.
Final 1 hr: Re-read the 12 solved questions above; cover the solutions and resolve mentally.
The drill plan pairs with the triangles cheatsheet for inscribed and circumscribed setups and the mensuration formulas for sector and segment area questions. The broader geometry cluster includes the coordinate geometry guide, which solves a subset of circle problems through the algebraic route. Aspirants targeting full CAT 2026 preparation can access the structured roadmap via the CAT 2026 waitlist.
- Read the figure for trigger phrases first; identify the theorem before reaching for a formula.
- Perpendicular from the centre to a chord bisects the chord; build the half-chord right triangle immediately.
- Two tangents from a single external point are equal; never re-derive this through congruent triangles.
- Intersecting chords, secant-tangent, and two-secant setups all collapse to the same power-of-a-point product rule.
- Angle in a semicircle is 90 degrees; whenever a diameter is mentioned, look for the right triangle it creates.
- Central angle equals twice the inscribed angle only when both vertices see the same arc; confirm the arc before doubling.
Circles on CAT is theorem-recognition first, computation second. 8 properties. 4 setups. 90-second solves.
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