Sec 3 E-Math · Circle properties

Equal chords questions with answers

Practice questions on equal chords, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

In the diagram, $AB$ and $CD$ are chords of the circle with centre $O$ . $OM$ is perpendicular to $AB$ and $ON$ is perpendicular to $CD$ . Both chords are $8$ cm long. Find $x$ .

Step 1 of 1

Find ON

ON = OM

(equal chords are equidistant from centre)

x = 2

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Questions

Question 1Equal chords: the distance from the centre
In the diagram, $AB$ and $CD$ are chords of the circle with centre $O$ . $OM$ is perpendicular to $AB$ and $ON$ is perpendicular to $CD$ . Both chords are $14$ cm long. Find $x$ .
Show the worked solution
1. Step 1: Find ON $ON = OM$ (equal chords are equidistant from centre) $x = 5$
Question 2Equal distances: the length of a chord
In the diagram, $AB$ and $CD$ are chords of the circle with centre $O$ . $OM$ is perpendicular to $AB$ and $ON$ is perpendicular to $CD$ . Find the length of the chord $CD$ , correct to 3 significant figures.
Show the worked solution
1. Step 1: Compare the chords AB and CD $AB = CD$ (chords equidistant from centre are equal)So finding $CD$ gives $AB$ too.
2. Step 2: OC and OA are both radii $OC = OA$ (radii) $OC = 11$ cm
3. Step 3: Pythagoras' theorem in triangle OCN $CN^2 = OC^2 - ON^2$ $CN^2 = 11^2 - 5^2$ $CN^2 = 96$ $CN = 9.7979\ldots$ cm
4. Step 4: Find CD $CD = 2 \times CN$ (⊥ from centre bisects chord) $CD = 19.5959\ldots$ $\phantom{CD} = 19.6$ cm (3 s.f.)
Question 3Equal chords: find the radius
In the diagram, $AB$ and $CD$ are chords of the circle with centre $O$ . $OM$ is perpendicular to $AB$ and $ON$ is perpendicular to $CD$ . Both chords are $16$ cm long. Find $x$ , correct to 3 significant figures.
Show the worked solution
1. Step 1: Find OM $OM = ON$ (equal chords are equidistant from centre) $OM = 5$ cm
2. Step 2: Find AM $AM = \dfrac{1}{2} AB$ (⊥ from centre bisects chord) $AM = 8$ cm
3. Step 3: Pythagoras' theorem in triangle OAM $OA^2 = OM^2 + AM^2$ $x^2 = 5^2 + 8^2$ $x^2 = 89$ $x = \sqrt{89}$ $x = 9.4339\ldots$ $\phantom{x} = 9.43$ cm (3 s.f.)
Question 4Equal chords: find an angle
In the diagram, $AB$ and $CD$ are chords of the circle with centre $O$ , and $AB = CD$ . Find $x$ .
Show the worked solution
1. Step 1: Find ∠OBA, then ∠AOB $\angle OBA = 52^\circ$ (base ∠s of isos. △) $\mathstrut \angle AOB = 180^\circ - 2 \times 52^\circ$ (∠ sum of △) $\angle AOB = 76^\circ$
2. Step 2: Compare triangles OAB and OCD (OA = OC and OB = OD are radii) $\triangle OAB \equiv \triangle OCD$ (SSS) $\angle COD = \angle AOB$ $x = 76$

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Worked example

In the diagram,

AB

and

CD

are chords of the circle with centre

O

OM

is perpendicular to

AB

and

ON

is perpendicular to

CD

. Both chords are

8

cm long. Find

x

Step 1 of 1

Find ON

ON = OM

(equal chords are equidistant from centre)

x = 2

Questions

Question 1Equal chords: the distance from the centre

In the diagram, $AB$ and $CD$ are chords of the circle with centre $O$ . $OM$ is perpendicular to $AB$ and $ON$ is perpendicular to $CD$ . Both chords are $14$ cm long. Find $x$ .

Show the worked solution

Step 1: Find ON $ON = OM$ (equal chords are equidistant from centre) $x = 5$

Question 2Equal distances: the length of a chord

In the diagram, $AB$ and $CD$ are chords of the circle with centre $O$ . $OM$ is perpendicular to $AB$ and $ON$ is perpendicular to $CD$ . Find the length of the chord $CD$ , correct to 3 significant figures.

Show the worked solution

Step 1: Compare the chords AB and CD $AB = CD$ (chords equidistant from centre are equal)So finding $CD$ gives $AB$ too.
Step 2: OC and OA are both radii $OC = OA$ (radii) $OC = 11$ cm
Step 3: Pythagoras' theorem in triangle OCN $CN^2 = OC^2 - ON^2$ $CN^2 = 11^2 - 5^2$ $CN^2 = 96$ $CN = 9.7979\ldots$ cm
Step 4: Find CD $CD = 2 \times CN$ (⊥ from centre bisects chord) $CD = 19.5959\ldots$ $\phantom{CD} = 19.6$ cm (3 s.f.)

Question 3Equal chords: find the radius

In the diagram, $AB$ and $CD$ are chords of the circle with centre $O$ . $OM$ is perpendicular to $AB$ and $ON$ is perpendicular to $CD$ . Both chords are $16$ cm long. Find $x$ , correct to 3 significant figures.

Show the worked solution

Step 1: Find OM $OM = ON$ (equal chords are equidistant from centre) $OM = 5$ cm
Step 2: Find AM $AM = \dfrac{1}{2} AB$ (⊥ from centre bisects chord) $AM = 8$ cm
Step 3: Pythagoras' theorem in triangle OAM $OA^2 = OM^2 + AM^2$ $x^2 = 5^2 + 8^2$ $x^2 = 89$ $x = \sqrt{89}$ $x = 9.4339\ldots$ $\phantom{x} = 9.43$ cm (3 s.f.)

Question 4Equal chords: find an angle

In the diagram, $AB$ and $CD$ are chords of the circle with centre $O$ , and $AB = CD$ . Find $x$ .

Show the worked solution

Step 1: Find ∠OBA, then ∠AOB $\angle OBA = 52^\circ$ (base ∠s of isos. △) $\mathstrut \angle AOB = 180^\circ - 2 \times 52^\circ$ (∠ sum of △) $\angle AOB = 76^\circ$
Step 2: Compare triangles OAB and OCD (OA = OC and OB = OD are radii) $\triangle OAB \equiv \triangle OCD$ (SSS) $\angle COD = \angle AOB$ $x = 76$