Chords Chords are line segments which have their $2$ end points on the circumference of a circle. In the circle below, $AB$ is a chord but $CD$ is not. Diameter which passes through the centre of the circle is the longest chord. Perpendicular Bisector of a Chord Circle Symmetrical Property $1$: • If a line $OM$ passes through the centre of a circle and is perpendicular to a chord $AB$ (which is not the diameter) of the circle, then the line $OM$ bisects the chord $AB$. • If a line $OM$ passes through the centre of a circle and bisects the chord $AB$ (which is not the diameter) of the circle, then the line $OM$ is perpendicular to a chord $AB$. • If a line $OM$ bisects the chord $AB$ and is perpendicular to the chord $AB$ (i.e. $OM$ is the perpendicular bisector of the chord $AB$), then the line $OM$ passes through the centre of the circle. (This is also true if the chord is a diameter of the circle.) Equal Chords Circle Symmetrical Property $2$: • Equal chords of a circle are equidistant from the centre of the circle. • Chords that are equidistant from the centre of a circle are equal (in length). Radius of a Circle and Tangent to a circle • A straight line cutting a circle at two distinct points is called a secant. $AB$ is a secant. • If a straight line and a circle have only one point of contact, then that line is called a tangent $CD$ is the tangent to the circle at only one point $A$. $EF$ is the tangent to the circle at only one point $B$. Circle Symmetrical Property $3$: The tangent at the point of contact is perpendicular to the radius of the circle. (i.e. $TAN \perp OA$) Circle Symmetrical Property $4$: • Tangents from an external point are equal (in length). • The line from the centre of a circle to an external point bisects the angle between the two tangents from the external point. Triangle $OBT$ and Triangle $OAT$ are congruent.

Symmetrical Properties of Circle

Secondary 3

Chords
Chords are line segments which have their $2$ end points on the circumference of a circle.
In the circle below, $AB$ is a chord but $CD$ is not.
Diameter which passes through the centre of the circle is the longest chord.
Perpendicular Bisector of a Chord
Circle Symmetrical Property $1$:
• If a line $OM$ passes through the centre of a circle and is perpendicular to a chord $AB$ (which is not the diameter) of the circle, then the line $OM$ bisects the chord $AB$.
• If a line $OM$ passes through the centre of a circle and bisects the chord $AB$ (which is not the diameter) of the circle, then the line $OM$ is perpendicular to a chord $AB$.
• If a line $OM$ bisects the chord $AB$ and is perpendicular to the chord $AB$ (i.e. $OM$ is the perpendicular bisector of the chord $AB$), then the line $OM$ passes through the centre of the circle. (This is also true if the chord is a diameter of the circle.)
Equal Chords
Circle Symmetrical Property $2$:
• Equal chords of a circle are equidistant from the centre of the circle.
• Chords that are equidistant from the centre of a circle are equal (in length).
Radius of a Circle and Tangent to a circle
• A straight line cutting a circle at two distinct points is called a secant.

$AB$ is a secant.

• If a straight line and a circle have only one point of contact, then that line is called a tangent

$CD$ is the tangent to the circle at only one point $A$.

$EF$ is the tangent to the circle at only one point $B$.

Circle Symmetrical Property $3$:
The tangent at the point of contact is perpendicular to the radius of the circle. (i.e. $TAN \perp OA$)
Circle Symmetrical Property $4$:
• Tangents from an external point are equal (in length).
• The line from the centre of a circle to an external point bisects the angle between the two tangents from the external point.
Triangle $OBT$ and Triangle $OAT$ are congruent.

Video solution locked

A free account saves your progress; this video needs a course plan.

680+ questions here come with free worked solutions. A free account saves your questions and progress; this one needs a course plan.

Create a free account to practise similar questions.

Need help? Join our E-Maths tuition for O-Level success.