The idea, step by step
Centre and chord
A perpendicular from a circle’s centre to a chord bisects that chord. This creates two right triangles with a common centre-to-chord distance and equal radii.
Conversely, for a non-diameter chord, the line from the centre to the chord midpoint is perpendicular to the chord. Check that the centre really is specified; an arbitrary interior point does not have these properties.
Tangents meet radii at right angles
A tangent touches the circle at one point. The radius to that point is perpendicular to the tangent, so Pythagoras can connect the radius, tangent length and centre-to-exterior-point distance.
Two tangent segments drawn from the same exterior point to one circle have equal lengths. This does not say that tangents from different exterior points are equal.
A common trap
Using half the chord as the radius.
Instead: A chord’s half-length and the centre-to-chord distance are the legs of a right triangle; the radius is its hypotenuse.
See the thinking
Distance to a chord
A circle has centre O, radius 5 cm and chord AB of length 8 cm. M is the foot of the perpendicular from O to AB. Find OM.
- The perpendicular bisects the chord, so AM=4 cm.
- In right triangle OMA, OM²=5²−4²=9.
- OM=3 cm.
Tangent length
PA is tangent at A. The circle centre is O, OA=5 cm and OP=13 cm. Find PA.
- OA⊥PA, so OAP is right-angled at A.
- PA²=13²−5²=144.
- PA=12 cm.
Check your understanding
Two tangent segments PA and PB from the same external point touch one circle. PA=7 cm. Find PB.
Show answer
7 cm
Tangents from the same exterior point have equal lengths.
Your turn
6 questions · 13 marks · Work at your own pace.
Read the questions online
- A radius 13 cm circle has chord length 10 cm. Find the perpendicular distance from centre to chord.2 marks
- A chord is 12 cm long and 3 cm from the centre. Find the radius.2 marks
- A tangent from P has length 15 cm and the radius to its contact point is 8 cm. Find the centre-to-P distance.2 marks
- Tangents PA and PB from P have lengths 2 x+1 and 3 x−4. Find x and the common tangent length.2 marks
- A radius 10 cm circle has a chord 16 cm long. A student uses 10²−16². Explain the error and give the correct distance.3 marks
- A line joins the circle centre to the midpoint of a non-diameter chord. What angle does it make with the chord?2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- A radius 13 cm circle has chord length 10 cm. Find the perpendicular distance from centre to chord.2 marks
Half-chord 5 cm.
Distance=√(169−25)=12.
12 cm - A chord is 12 cm long and 3 cm from the centre. Find the radius.2 marks
.
cm - A tangent from P has length 15 cm and the radius to its contact point is 8 cm. Find the centre-to-P distance.2 marks
√(15²+8²)=17.
17 cm - Tangents PA and PB from P have lengths 2 x+1 and 3 x−4. Find x and the common tangent length.2 marks
2 x+1=3 x−4 gives x=5.
2×5+1=11.
x=5; length 11 - A radius 10 cm circle has a chord 16 cm long. A student uses 10²−16². Explain the error and give the correct distance.3 marks
The right-triangle leg is 8, not 16.
√(100−64)=6.
Use half the chord; distance 6 cm. - A line joins the circle centre to the midpoint of a non-diameter chord. What angle does it make with the chord?2 marks
The point must be the circle’s centre and the other point the chord midpoint.
90°.
Curriculum reference & lesson scope
Australian Curriculum v9 and SA Curriculum
Angle and similarity reasoning; congruent/similar triangles in proofs; chords, radii and tangents; angles at the centre and circumference; circle theorems and their applications; connected deductive arguments; construction/spatial algorithms; testing and justifying solutions.
Official curriculum source ↗