Geometry Transformations and Similarity

Congruent and Similar Shapes

In geometry, we classify the relationship between two shapes based on how their properties compare. Two shapes are congruent if they are identical in both shape and size. This means all corresponding sides are equal in length and all corresponding angles are equal in magnitude. Congruent shapes are essentially the same shape in different positions or orientations.

Two shapes are similar if they have the same shape but are not necessarily the same size. For two shapes to be similar, all corresponding angles must be equal, and the lengths of all corresponding sides must be in the same proportion. This proportion is known as the scale factor.

Transformations are the mechanisms we use to move or change a shape (the pre-image) into another (the image). Rigid transformations, which include translations, rotations, and reflections, result in congruent images. Non-rigid transformations, specifically enlargements, result in similar images.

Translations as Vectors

A translation moves every point of a shape by the same distance in a specified direction. In the TMUA, we describe translations using 2-dimensional vectors. A vector is written in the form $\begin{pmatrix} x \\ y \end{pmatrix}$, where $x$ represents the horizontal displacement and $y$ represents the vertical displacement.

Example

If a triangle with vertices $(1, 2)$, $(3, 2)$, and $(1, 5)$ is translated by the vector $\begin{pmatrix} 4 \\ -3 \end{pmatrix}$, we find the new vertices by adding the vector components to the coordinates of each vertex:

1. $(1+4, 2-3) = (5, -1)$
2. $(3+4, 2-3) = (7, -1)$
3. $(1+4, 5-3) = (5, 2)$

The resulting image is congruent to the original.

Reflections and Rotations

Reflections involve flipping a shape over a line of reflection. Every point in the image is the same perpendicular distance from the line of reflection as the corresponding point in the pre-image. Common lines of reflection on coordinate axes include $x = a$, $y = b$, $y = x$, and $y = -x$. Note that reflections reverse the orientation of the shape (e.g., a clockwise sequence of vertices becomes anticlockwise).

Rotations turn a shape around a fixed point called the centre of rotation. To fully describe a rotation, you must specify the centre of rotation, the angle of rotation, and the direction (clockwise or anticlockwise). Unless stated otherwise, anticlockwise is often treated as the positive direction in higher mathematics, but TMUA questions will typically be explicit.

Exercise

Consider the point $P(2, 5)$. Find the image of $P$ after a reflection in the line $y = x$, followed by a rotation of $90^{\circ}$ clockwise about the origin $(0, 0)$.

Enlargements

An enlargement changes the size of a shape while keeping its angles and proportions the same, thus creating a similar shape. It is defined by a centre of enlargement and a scale factor $k$. To find the image of a point, we measure the vector from the centre to the point and multiply it by $k$.

1. If $k > 1$, the shape is enlarged and moved further from the centre.
2. If $0 < k < 1$, the shape is reduced in size (fractional scale factor) and moved closer to the centre.
3. If $k < 0$ (negative scale factor), the shape is inverted and appears on the opposite side of the centre of enlargement.

Example

A square has vertices $(1, 1), (2, 1), (2, 2),$ and $(1, 2)$. It is enlarged by a scale factor of $-2$ with the centre of enlargement at the origin $(0, 0)$. The new vertices are found by multiplying each coordinate by $-2$:

$(-2, -2), (-4, -2), (-4, -4),$ and $(-2, -4)$.

The image is twice as large as the original but is inverted through the origin.

Combinations and Invariance

When multiple transformations are applied in sequence, the order often matters. A combination of transformations can sometimes be described as a single transformation of a different type. For instance, reflecting a shape in two parallel lines is equivalent to a single translation, while reflecting in two intersecting lines is equivalent to a rotation about the point of intersection.

Invariance refers to properties or points that remain unchanged by a transformation.

• An invariant point is a point that is mapped onto itself. In a rotation, the centre of rotation is the only invariant point. In a reflection, every point on the line of reflection is invariant.
• An invariant line is a line where every point on the line is mapped to another point on the same line. For example, in an enlargement, any line passing through the centre of enlargement is an invariant line, even though the individual points on that line move.