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SPI Mathematics · Circles

Chords, Angles and Cyclic Quadrilaterals

The perpendicular from the centre cuts a chord in half, the angle at the centre is twice the angle at the circle, and opposite angles of a cyclic quadrilateral add to 180 degrees.

All Circles notes

Why this matters

Ten questions in the bank, including both circle questions from the 2022 paper and the chapter's only two HARD ones. Chord lengths are one right triangle again; the angle questions need the right theorem picked from a figure, which is why they run harder.

Concept 1 of 3: The perpendicular from the centre bisects a chord

Drop a perpendicular from the centre to a chord. It lands on the middle of the chord, and the radius to one end of the chord makes a right triangle with half the chord and the distance from the centre.

Definition

  • The perpendicular from the centre to a chord cuts it in half.
  • With radius rr, distance dd from the centre and half chord hh: r2=d2+h2r^2 = d^2 + h^2.
  • The longest chord is the diameter (d=0d = 0). A longer chord is nearer the centre, and equal chords are equally far from it.
  • Two circles with the same centre are concentric. A chord of the larger circle that touches the smaller one is a tangent to the smaller circle, so the small radius is the distance dd and the point of contact is the middle of the chord.

Chord length

chord=2r2−d2r2=d2+(chord2)2\text{chord} = 2\sqrt{r^2 - d^2} \qquad r^2 = d^2 + \left(\tfrac{\text{chord}}{2}\right)^2
  • rradius of the circle
  • ddistance of the chord from the centre

Worked example

A circle has radius 10 cm. Find the length of a chord that is 6 cm from the centre.
Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Circles · Chords, Angles and Cyclic Quadrilaterals

A chord is at a distance of 8 cm from the centre of a circle of radius 17 cm. The length of the chord is

Stopping at the half chord

Pythagoras gives half the chord. With radius 10 cm and distance 6 cm the half chord is 8 cm, and the chord is 16 cm.

Putting the whole chord in the right triangle

The right triangle has the radius, the distance from the centre and HALF the chord. Halve the chord before you square it.
Drill 3 more on the perpendicular from the centre bisects a chord

Concept 2 of 3: Angles made by an arc

An arc can be seen from the centre or from a point on the circle. From the centre it looks twice as wide, and from every point on the same side of the circle it looks equally wide.

Definition

  • The angle an arc makes at the centre is twice the angle it makes at any point on the rest of the circle.
  • Angles in the same segment are equal: points on the same side of a chord see it at the same angle.
  • The angle in a semicircle is 90∘90^\circ: a diameter's ends and any other point on the circle make a right angle.
  • So a circle through all four corners of a rectangle has its diagonal as a diameter, and the circle through the corners of a right triangle has radius half the hypotenuse.

Angles in a circle

∠AOB=2 ∠ACBangle in a semicircle=90∘\angle AOB = 2\,\angle ACB \qquad \text{angle in a semicircle} = 90^\circ

Worked example

A, B and C lie on a circle with centre O, with C on the major arc. If ∠AOB=110∘\angle AOB = 110^\circ, find ∠ACB\angle ACB.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SPI · 2022 · SPI Entrance Examination 2022 · Q63Moderate

Example 2 · Circles · Chords, Angles and Cyclic Quadrilaterals

In the figure, m∠ACDm\angle ACD= 65deg then m∠CBDm\angle CBD= ____deg.

Halving when you should double

The angle at the centre is the larger one. If the angle at the circle is 64∘64^\circ, the angle at the centre is 128∘128^\circ, not 32∘32^\circ.

Using an angle from the other side of the chord

Only points on the same side of a chord see it at equal angles. Two points on opposite sides make angles that add to 180∘180^\circ.
Drill 2 more on angles made by an arc

Concept 3 of 3: Opposite angles of a cyclic quadrilateral

When all four corners of a quadrilateral lie on a circle, each pair of opposite angles stands on arcs that together make the whole circle, so the pair adds to 180 degrees.

Definition

  • A quadrilateral with all four corners on a circle is cyclic.
  • Opposite angles add to 180∘180^\circ: ∠A+∠C=180∘\angle A + \angle C = 180^\circ and ∠B+∠D=180∘\angle B + \angle D = 180^\circ.
  • If one side is extended, the exterior angle equals the interior opposite angle.
  • A parallelogram can be cyclic only if it is a rectangle, because its opposite angles are equal and must add to 180∘180^\circ.

Cyclic quadrilateral

∠A+∠C=180∘∠B+∠D=180∘\angle A + \angle C = 180^\circ \qquad \angle B + \angle D = 180^\circ

Worked example

ABCD is a cyclic quadrilateral with ∠A=(3x+4)∘\angle A = (3x + 4)^\circ and ∠C=(2x+6)∘\angle C = (2x + 6)^\circ. Find ∠A\angle A.
Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Circles · Chords, Angles and Cyclic Quadrilaterals

In a cyclic quadrilateral ABCD, it is being given that∠A=(x+y+10)∘\angle A = (x + y + 10)^\circ ∠B=(y+20)∘\angle B = (y + 20)^\circ, ∠C=(x+y−30)∘\angle C = (x + y - 30)^\circ and ∠D=(x+y)∘\angle D = (x + y)^\circ. Then, ∠B\angle B = ?

Adding neighbouring angles to 180

Only opposite angles of a cyclic quadrilateral add to 180∘180^\circ. Angles A and B are neighbours, and their sum can be anything.

Matching the exterior angle with the angle next to it

The exterior angle at C equals the interior angle at A, the opposite corner. The interior angle at C, right next to it, is the one it adds to 180∘180^\circ with.
Drill 2 more on opposite angles of a cyclic quadrilateral

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

  • The perpendicular from the centre bisects a chord

    Chord length

    chord=2r2−d2r2=d2+(chord2)2\text{chord} = 2\sqrt{r^2 - d^2} \qquad r^2 = d^2 + \left(\tfrac{\text{chord}}{2}\right)^2
  • Angles made by an arc

    Angles in a circle

    ∠AOB=2 ∠ACBangle in a semicircle=90∘\angle AOB = 2\,\angle ACB \qquad \text{angle in a semicircle} = 90^\circ
  • Opposite angles of a cyclic quadrilateral

    Cyclic quadrilateral

    ∠A+∠C=180∘∠B+∠D=180∘\angle A + \angle C = 180^\circ \qquad \angle B + \angle D = 180^\circ

Watch out for (6)

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