CDS Mathematics · Circles

Chords and the Perpendicular from the Centre

The perpendicular from the centre bisects a chord, so the radius, the half-chord and the distance from the centre make a right triangle.

All Circles notes CDS Mathematics strategy

Why this matters

Fifteen PYQs, five of them HARD. Almost every one is the same right triangle: radius as hypotenuse, half the chord and the distance from the centre as the legs. Parallel-chord questions add one choice — same side or opposite sides of the centre.

Concept 1 of 2: Half-chord, distance and radius

Drop a perpendicular from the centre to a chord. It lands on the chord's midpoint, and the radius to either end closes a right triangle. Any two of the three lengths give the third.

Definition

The perpendicular from the centre bisects the chord, and the line from the centre to a chord's midpoint is perpendicular to it.
A chord of length $c$ at distance $d$ from the centre: $r^2 = d^2 + \left(\dfrac c2\right)^2$ .
Equal chords are equally far from the centre; a longer chord is nearer.
Two parallel chords at distances $d_1, d_2$ : $|d_1 - d_2|$ apart on the same side of the centre, $d_1 + d_2$ on opposite sides.
A chord subtends $\theta$ at the centre when $\sin\dfrac\theta2 = \dfrac{c/2}{r}$ .

Chord and distance

r^2 = d^2 + \left(\dfrac{c}{2}\right)^2

Worked example

In a circle of radius

17

cm, parallel chords are

16

cm and

30

cm long. How far apart are they?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2019 · CDS (II) 2019 — Elementary Mathematics · Q74Moderate

Example 1 · Circles · Chords and Perpendiculars

If the lengths of two parallel chords in a circle of radius 10 cm are 12 cm and 16 cm, then what is the distance between these two chords ?

Parallel chords have two answers

Unless the question says which side of the centre each chord lies on, both the difference and the sum of the distances are possible. The options often pair them, as in '2 cm or 14 cm'.

Half the chord, not the chord

The leg of the right triangle is HALF the chord. A

24

cm chord in a circle of radius

13

5

cm from the centre, not

\sqrt{24^2 - 13^2}

Drill 9 more on half-chord, distance and radius

Concept 2 of 2: Arches, segment heights and equal chords

The height of a segment is measured from the chord's midpoint up to the arc. The centre is

r - h

beyond the chord, so the same right triangle gives an equation in

r

alone.

Definition

A chord of length $c$ with segment height $h$ : $r^2 = \left(\dfrac c2\right)^2 + (r - h)^2$ , so $r = \dfrac{(c/2)^2 + h^2}{2h}$ .
An arch of span $c$ and height $h$ is the same problem.
The midpoint of an arc, the midpoint of its chord and the centre lie on one line.
Two equal chords $AB = AC = a$ from one point: $BC = \dfrac{a}{r}\sqrt{4r^2 - a^2}$ .

Radius from a segment

r = \dfrac{(c/2)^2 + h^2}{2h}

Worked example

A chord

24

cm long cuts off a segment of height

6

cm. Find the radius.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2023 · CDS (II) 2023 — Elementary Mathematics · Q62Moderate

Example 2 · Circles · Chords and Perpendiculars

The arch of a bridge is in the form of an arc of a circle. If the span of the bridge is 40 m and height in the middle is 8 m, then what is the radius of curvature of the bridge ?

The centre is r − h from the chord

The height is measured from the chord to the arc, so the centre sits

r - h

from the chord, not

h

. Using

h

makes the leg the wrong side of the triangle.

Radius or diameter?

Segment questions often ask for the DIAMETER. The equation gives

r

; double it before matching options.

Drill 4 more on arches, segment heights and equal chords

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 (2)

Half-chord, distance and radius
Chord and distance
$r^2 = d^2 + \left(\dfrac{c}{2}\right)^2$
Arches, segment heights and equal chords
Radius from a segment
$r = \dfrac{(c/2)^2 + h^2}{2h}$

Watch out for (4)

Parallel chords have two answers
→ Half-chord, distance and radius
Half the chord, not the chord
→ Half-chord, distance and radius
The centre is r − h from the chord
→ Arches, segment heights and equal chords
Radius or diameter?
→ Arches, segment heights and equal chords

Test yourself on Circles

15 past CDS questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Circles chapter test Drill all 15 questions