Mathematics
Symmetry
19. Symmetry
LEARNING OBJECTIVES
Overview
1. Identify line symmetry in 2D shapes
2. Find the order of rotational symmetry of 2D shapes
3. Recognise and use symmetrical properties of triangles,
quadrilaterals and circles
4. Recognise symmetry properties of prisms and pyramids
5. APPLY symmetry properties of circles to solve probles
19. Symmetry
RECAP
2D shapes are symmetrical
if they can be divided into two identical halves by a
straight line.
Each half is the mirror image of the other.
19 Symmetry
19. Symmetry
RECAP
3D solids are symmetrical.
If they can be ‘cut’ into two identical parts by a plane.
The two parts are mirror images of each other.
Each half is the mirror image of the other.
19 Symmetry
19. Symmetry
2D symmetry
19.1 2D symmetry
1. Line symmetry
2. Rotational symmetry
19. Symmetry
2D symmetry
1. Line symmetry
Shapes need to be mirror images to be symmetrical
Shapes can have more than one line of symmetry
19.1 2D symmetry
19. Symmetry
2D symmetry
2.Rotational symmetry
Rotate a shape through 360 degrees and keep the
centre point fixed. How many times does it fit?
19.1 2D symmetry
This video explains it clearly!
https://www.youtube.com/watch?v=nt43FJQppCQ&t=35s
19. Symmetry
3D symmetry
2 types of symmetry in 3D shapes
19.2 Symmetry in
3D shapes
1. Plane symmetry
2. Rotational symmetry
19. Symmetry
3D symmetry
1. Plane symmetry
Plane: flat surface (cut the solid in half and see the
mirror image)
Can have more than one plane
19.2 Symmetry in
3D shapes
19. Symmetry
3D symmetry
2. Rotational symmetry
Contains “axis”
Looks the same when you rotate it about the axis
19.2 Symmetry in
3D shapes
19. Symmetry
Symmetry in circles
Circles have lines of symmetry and rotational symmetry
about its center.
Because of these, the following facts, a number of results
can be concluded:
19.3 Symmetry
properties of
circles
1. The perpendicular bisector of a chord passes through
the centre.
2. Equal chords are equidistant from the centre, and
chords equidistant from the centre are equal in length.
3. Two tangents drawn to a circle from the same point
outside the circle are equal in length.
19. Symmetry
Symmetry in circles
19.3 Symmetry
properties of
circles
1. The perpendicular bisector of a chord passes through
the center.
The perpendicular from the centre of a circle to a
chord meets the chord at its mid-point.
The line joining the centre of a circle to the mid-point
of a chord is perpendicular to the chord
19. Symmetry
Symmetry in circles
2. Equal chords are equidistant from the center and are
therefore equal in length
19.3 Symmetry
properties of
circles
19. Symmetry
Symmetry in circles
19.3 Symmetry
properties of
circles
3. Two tangents drawn to a circle from the same point
outside the circle are equal in length
the tangents subtend equal angles at the center (i.e.
angle POA = angle POB)
the line joining the center to the point where the
tangents meet bisects the angle between the tangents.
(i.e. angle OPA = angle OPB)
19. Symmetry
Symmetry in circles
19.4 Angle
relationships in
circles
1. Angle in a semi-circle
2. Angle between the tangent and radius
3. Angle at the center of a circle is twice the angle at the
circumference
4. Angle in the same segment are equal
5. Opposite angles of a cyclic quadrilateral
6. Each exterior angle of a cyclic quadrilateral = interior
angle opposite to it
19. Symmetry
Symmetry in circles
19.4 Angle
relationships in
circles
1. Angle in a semi-circle
2. Angle between the tangent and radius
3. Angle at the center of a circle is twice the angle at the
circumference
4. Angle in the same segment are equal
5. Opposite angles of a cyclic quadrilateral
6. Each exterior angle of a cyclic quadrilateral = interior
angle opposite to it
19. Symmetry
Symmetry in circles
19.4 Angle
relationships in
circles
1. Angle in a semi-circle
2. Angle between the tangent and radius
3. Angle at the center of a circle is twice the angle at the
circumference
4. Angle in the same segment are equal
5. Opposite angles of a cyclic quadrilateral
6. Each exterior angle of a cyclic quadrilateral = interior
angle opposite to it
19. Symmetry
Symmetry in circles
19.4 Angle
relationships in
circles
1. Angle in a semi-circle
2. Angle between the tangent and radius
3. Angle at the center of a circle is twice the angle at the
circumference
4. Angle in the same segment are equal
5. Opposite angles of a cyclic quadrilateral
6. Each exterior angle of a cyclic quadrilateral = interior
angle opposite to it
19. Symmetry
Symmetry in circles
19.4 Angle
relationships in
circles
1. Angle in a semi-circle
2. Angle between the tangent and radius
3. Angle at the center of a circle is twice the angle at the
circumference
4. Angle in the same segment are equal
5. Opposite angles of a cyclic quadrilateral
6. Each exterior angle of a cyclic quadrilateral = interior
angle opposite to it
19. Symmetry
Symmetry in circles
19.4 Angle
relationships in
circles
1. Angle in a semi-circle
2. Angle between the tangent and radius
3. Angle at the center of a circle is twice the angle at the
circumference
4. Angle in the same segment are equal
5. Opposite angles of a cyclic quadrilateral
6. Each exterior angle of a cyclic quadrilateral = interior
angle opposite to it