Circles: Chords and Tangents
Geometry · Circles: Chords and Tangents
Syllabus tag: KCSE | Mathematics | Form 3 | Topic 7 Circles: Chords and Tangents
Lesson objectives
By the end of this topic, you should be able to:
- Calculate the length of an arc and of a chord.
- Calculate lengths of tangents and intersecting chords.
- State and use the properties of chords.
- Apply the alternate segment theorem and the properties of touching circles.
Circles: Chords and Tangents
This topic collects the length and angle results for chords and tangents.
a) Arc and chord length
Arc length = (θ/360) × 2πr, where θ is the angle at the centre.
A chord is the straight line joining the arc's ends.
b) Properties of chords
A perpendicular from the centre to a chord bisects it. That is the key result, and almost every chord calculation uses it.
The converse holds too: the line from the centre to a chord's midpoint is perpendicular to it.
Equal chords are equidistant from the centre, and chords equidistant from the centre are equal.
The perpendicular bisector of any chord passes through the centre. That is how you find a circle's centre from three points on it.
c) Calculating chord length
The centre, the chord's midpoint and one end form a right-angled triangle.
Its hypotenuse is the radius. One leg is the distance from centre to chord, and the other is half the chord.
Apply Pythagoras, then remember to double the half-chord. Forgetting to double is the usual slip.
d) Tangents
A tangent touches the circle at exactly one point.
A tangent is perpendicular to the radius at the point of contact. Every tangent calculation starts there.
Two tangents from an external point are equal in length.
The line from that external point to the centre bisects the angle between the tangents. It also bisects the chord joining the two contact points.
e) The alternate segment theorem
The angle between a tangent and a chord equals the angle in the alternate segment.
The alternate segment lies on the other side of the chord from the angle you are measuring.
Suppose a tangent and chord make an angle of 55°. Any angle drawn in the segment beyond that chord is also 55°.
Identify the chord first, then look across it. Choosing the wrong segment is the usual error.
f) Intersecting chords
When two chords cross inside a circle at P: AP × PB = CP × PD.
When two lines cross outside: PA × PB = PC × PD, measuring from the external point.
If one line is a tangent: PT² = PA × PB, since the two tangent lengths coincide.
g) Touching circles
Two circles touching externally have their centres a distance R + r apart.
Two circles touching internally have their centres R − r apart.
The point of contact lies on the line joining the centres, in both cases. That gives the right-angled triangles needed for calculation.
h) Where this is used
Belt and pulley problems. Arch and bridge design. Gear layouts. Any construction with circular parts meeting.
Words to know
- Tangent -- a line touching a circle at exactly one point.
- Point of contact -- where a tangent meets the circle.
- Secant -- a line cutting a circle at two points.
- Alternate segment -- the segment on the opposite side of a chord from a given angle.
- Common tangent -- a line touching two circles.
:::checkpoint Check yourself
- A chord is 8 cm from the centre of a circle of radius 17 cm. Find its length.
- What angle does a tangent make with the radius at the point of contact?
- State the alternate segment theorem.
- Two circles of radii 7 cm and 4 cm touch externally. How far apart are their centres? :::
Bridge to practice
The exercises begin with chord properties and the perpendicular from the centre, move through tangents and the intersecting chords theorem, and finish with the alternate segment theorem and touching circles. Whenever a tangent appears in a diagram, draw the radius to the point of contact before doing anything else.