Circles (Tangent and Cyclic Properties)

Chapter in a nutshell: This chapter combines the angle, chord, cyclic and tangent properties of circles. Key results: the angle at the centre is twice that at the circumference, angles in the same segment are equal, opposite angles of a cyclic quadrilateral are supplementary, a tangent is perpendicular to the radius, and two tangents from an external point are equal.

1. Angle Properties

• Angle at centre = 2 × angle at circumference standing on the same arc.
• Angles in the same segment are equal.
• Angle in a semicircle = 90° (angle subtended by a diameter).
• Equal arcs subtend equal angles (at the centre/circumference).

2. Chord Properties

• The perpendicular from the centre to a chord bisects the chord (and vice versa).
• Equal chords are equidistant from the centre (and vice versa).
• Intersecting chords: if two chords meet at P, $PA\times PB = PC\times PD$.

3. Cyclic Quadrilateral

• Opposite angles are supplementary: $\angle A+\angle C=180^\circ$, $\angle B+\angle D=180^\circ$.
• The exterior angle = interior opposite angle.

4. Tangent Properties

• A tangent is perpendicular to the radius at the point of contact.
• Two tangents from an external point are equal in length.
• Tangent–chord angle = angle in the alternate segment (alternate segment theorem).
• Tangent–secant: $PT^2 = PA\times PB$ (PT tangent, PAB secant from external point P).

5. Formulas (quick reference)

• Angle at centre $=2\times$ angle at circumference (same arc).
• Cyclic quad: opposite angles sum to $180^\circ$.
• Length of tangent from external point at distance $d$ from centre: $\sqrt{d^2-r^2}$.
• Intersecting chords: $PA\cdot PB=PC\cdot PD$; tangent–secant: $PT^2=PA\cdot PB$.

6. Prerequisite Formulas (earlier classes — in full)

• Circumference of a circle: $C=2\pi r=\pi d$.
• Area of a circle: $A=\pi r^2$.
• Diameter: $d=2r$.
• Length of an arc (angle $\theta$): $\ell=\dfrac{\theta}{360^\circ}\times2\pi r$.
• Area of a sector: $=\dfrac{\theta}{360^\circ}\times\pi r^2$.
• Area of a segment = area of sector − area of the triangle.
• Angle sum of a triangle $=180^\circ$; of a quadrilateral $=360^\circ$.
• Pythagoras theorem: $\text{hyp}^2=\text{base}^2+\text{perpendicular}^2$ (used with tangent ⟂ radius).
• Isosceles triangle: base angles equal (used since two tangents/radii are equal).

7. Worked Example (one, for the method)

8. Common Mistakes to Avoid

• Using "angle at centre = angle at circumference" — it is twice.
• Forgetting the angle in a semicircle is 90°.
• Saying adjacent (not opposite) angles of a cyclic quad are supplementary.
• Forgetting tangent ⟂ radius at the point of contact.
• Mixing the intersecting-chords result with the tangent–secant ($PT^2$) result.

9. Likely Exam Questions (with crisp answers)

1. Relation between central and inscribed angles on the same arc? → Central = 2 × inscribed.
2. Angle in a semicircle? → 90°.
3. Opposite angles of a cyclic quadrilateral sum to? → 180°.
4. Angle between a tangent and the radius at the point of contact? → 90°.
5. Lengths of two tangents from an external point? → Equal.
6. State the alternate segment theorem. → The tangent–chord angle equals the inscribed angle in the alternate segment.
7. Intersecting chords relation? → $PA\cdot PB=PC\cdot PD$.
8. Tangent–secant relation? → $PT^2=PA\cdot PB$.
9. Circumference and area formulas? → $2\pi r$ and $\pi r^2$.
10. Length of arc for angle $\theta$? → $\frac{\theta}{360}\times2\pi r$.
11. Area of a sector? → $\frac{\theta}{360}\times\pi r^2$.
12. What does the perpendicular from the centre to a chord do? → Bisects the chord.

Extended & Prerequisite Formulas (full)

• Central∠ = 2×inscribed∠ (same arc); semicircle∠=90°; same-segment∠ equal; cyclic quad opposite∠ sum 180°.
• Tangent⟂radius; equal tangents from external point; tangent length √(d²−r²); PA·PB=PC·PD (chords); PT²=PA·PB (tangent-secant).
• Prereq: circumference 2πr; area πr²; arc $\tfrac{\theta}{360}2\pi r$; sector $\tfrac{\theta}{360}\pi r^2$; segment = sector − triangle; Pythagoras.