# Symmetry

## Key Questions

It is a line about which the shape/curve is repeated as if it had rotated about it by ${180}^{o}$
See diagram/graph

#### Explanation:

$\textcolor{g r e e n}{\text{The line of symmetry is the y axis}}$

$f \left(x , y\right) = {x}^{2} + x y + {y}^{2}$
$g \left(x , y , z\right) = x y + y z + z x + \frac{1}{{x}^{2} + {y}^{2} + {z}^{2}}$

#### Explanation:

A symmetric function is a function in several variable which remains unchanged for any permutation of the variables.

For example, if $f \left(x , y\right) = {x}^{2} + x y + {y}^{2}$, then $f \left(y , x\right) = f \left(x , y\right)$ for all $x$ and $y$.

How many times is the same shape seen if a figure is turned through 360°

#### Explanation:

Symmetry means that there is a 'sameness' about two figures THere are two types of symmetry - line symmetry and rotational symmetry.

Line symmetry means if you draw a line thorugh the middle of a figure, the one side is a mirror image of the other.

Rotational symmetry is the symmetry of turning.

If you turn a shape though 360°, sometimes the identical shape is seen again during the turn. This is called rotational symmetry.

For example, a square has 4 sides, but the square will look exactly the same no matter which of its sides is at the top.

Rotational symmetry is described by the number of times the same shape is seen during the 360° rotation.

A square has rotational symmetry of order 4,
An equilateral triangle has rotational symmetry of order 3.
A rectangle and a rhombus have rotational symmetry of order 2.

A regular pentagon has rotational symmetry of order 5.