## Reflections: A type of transformation

### Understanding Reflection

**How to perform a Reflection:**

**Draw the Line of Reflection**: This line will act as the axis over which the shape is flipped e.g. if the line of reflection is y = 3, it runs horizontally across the grid at y = 3.**Find the corresponding points**: For each point on the original shape, measure the perpendicular distance to the line of reflection.**Plot the mirrored points**: Reflect each point across the line to a point equally distant on the other side. For example, if a point is 2 units above the line y = 3, its reflection will be 2 units below at y = 1.**Connect the points**: Join the reflected points to create the new shape. Make sure the original and the new shapes are identical.

**Example in detail**

- The vertex (1, 2) is 1 unit below y = 3. Its reflection will be at (1, 4), 1 unit above y = 3.
- The vertex (4, 2) follows the same rule and reflects to (4, 4).
- The vertex (2, 5) is 2 units above y = 3. Its reflection will be at (2, 1), 2 units below y = 3.
- We can join these new vertices to create the reflected triangle.

### 1

For a square that has corners at (2, 2), (2, 5), (5, 5) and (5, 2), which line would correctly reflect the square to appear upside down?

## Overview of Rotations in Maths

**centre of rotation**. The rotation transformation helps us understand motion in physical spaces and solve geometric problems in symmetry and orientation.

### Understanding Rotations

**How to perform a rotation:**

**Find the centre of rotation**: This point is the pivot around which the shape rotates. It can be a vertex of the shape, a point inside, or outside the shape.**Find the angle and direction of rotation**: Decide how much and in which direction the shape should rotate, such as 90° clockwise.**Rotate each point**: Using a protractor or a digital tool, measure the angle from each point of the shape to its new position.**Connect the points**: After moving all points, draw lines to connect them, creating the rotated shape.

**Detailed example:**

- Each vertex of the rectangle is moved along an arc of a circle centred at (2.5, 2.5), maintaining the distance to the centre.
- The vertex (1, 1) moves to (1, 4), (1, 4) moves to (4, 4), (4, 4) moves to (4, 1) and (4, 1) moves back to (1, 1).

### 2

If you rotate a triangle with vertices at (0,0), (4,0) and (4,3) around the origin by 180 degrees, what are the new coordinates?

## Understanding Translations in Mathematics

### What is a translation?

**How to perform a translation:**

**Find the translation vector**: This vector shows the direction and distance the shape should move. It's normally written in column vector form - e.g. (a over b) - where 'a' and 'b' represent the horizontal and vertical shifts.**Apply the vector to each point**: Add the vector to the coordinates of each vertex of the shape. For example, if a point is at (x, y) and the vector is (3 over -2), the translated point would be at (x+3, y-2).**Connect the translated points**: Draw lines connecting the new points to form the translated shape.

**Detailed example:**

- The vertex (2, 3) moves to (-1, 7), (5, 3) to (2, 7), (5, 6) to (2, 10) and (2, 6) to (-1, 10).
- This results in the quadrilateral being shifted 3 units left and 4 units up.

### 3

What vector would you use to move a triangle from (1, 1), (4, 1), (2, 4) to (3, 5), (6, 5), (4, 8)?

## Enlargements

### What is an enlargement?

**centre of enlargement**. This centre can be inside, outside, or even on the shape. The scale factor determines how much bigger or smaller the shape becomes. A scale factor greater than 1 means an increase in size, while a scale factor between 0 and 1 means a reduction.

**How to perform an enlargement:**

**Find the centre of enlargement and scale factor**: Choose the fixed point that the shape is scaled from on the tracing paper and determine the scale factor.**Connect points to the centre**: Draw lines from each vertex of the shape through the centre of enlargement.**Apply the scale factor**: Extend or reduce the lines created in the previous step by multiplying the distances from the centre to each vertex by the scale factor.**Mark the new points and connect them**: Decide the new positions of the vertices along the extended or reduced lines. Connect these points to make the enlarged or reduced shape.

**Detailed example:**

- The vertex (1, 2) remains fixed as it's the centre.
- The vertex (4, 2) moves to (7, 2), doubling the distance from the centre.
- The vertex (1, 5) moves to (1, 8), also doubling the distance vertically.
- These new points create a triangle twice the size of the original.

### 4

What happens to a rectangle with vertices at (3, 3), (6, 3), (6, 6) and (3, 6) if it's enlarged by a scale factor of 0.5 with the centre of enlargement at (3, 3)?

## Combined Transformations

### Understanding combined transformations

**How to perform combined transformations:**

**Identify the sequence of transformations**: Find out the order that transformations are applied to the shape.**Apply the first transformation**: Translate the shape according to its rules.**Apply all the following transformations**: Apply the next transformations to the shape in the correct order.**Check the final position and shape**: After all transformations have been applied, check the final position and orientation of the shape.

**Detailed example:**

- After translation, the square moves to (4,3), (4,6), (7,6), (7,3).
- Rotating it 90 degrees clockwise about the origin positions the vertices at (-3,4), (-6,4), (-6,7) and (-3,7).
- Reflecting across the line y = x swaps every x and y coordinate. This leads to the vertices at (4,-3), (4,-6), (7,-6) and (7,-3).

### 5

If a triangle with vertices at (2, 1), (5, 1) and (3, 4) is first reflected across the x-axis and then translated by (-2 over 3), what are the new coordinates of the vertices?

## Inverse Transformations in Mathematics

### Understanding inverse transformations

**How to perform inverse transformations:**

**Identify the original transformation**: Find the first transformation that was applied, such as the centre and angle of a rotation.**Apply the inverse rule**:**Translation**: Apply a translation using the opposite vector. For example, if the original vector was (3 over -2) on the graph, the inverse is (-3 over 2).**Reflection**: Reflect the shape across the same line of reflection again.**Rotation**: Rotate the shape in the opposite direction by the same angle around the same centre.**Enlargement**: Use the reciprocal of the scale factor about the same centre of enlargement.**Verify the outcome**: Check the shape has returned to its original state to make sure you have applied the inverse transformation correctly.

**Detailed example:**

- Invert the rotation by 180° about the same point (5,5), which brings the shape back to its post-translation position.
- Translate the rectangle by (-4 over 3) to return to the original coordinates.

### 6

A shape is reflected across the line y = x and enlarged by a factor of 2 centred at the origin. What sequence of inverse transformations will restore the original shape?

## The properties of transformations in mathematics

### Key Properties of Transformations

**Preservation of distance**: Most basic transformations are isometric, such as translations, rotations and reflections. They maintain the distances between points in a shape to keep the size and proportions of the shape the same when it is transformed.**Preservation of angle**: Rotations, reflections and translations maintain the angles inside shapes. This keeps the geometric integrity of figures during transformations.**Orientation changes**: Some transformations change the orientation of shapes. A reflection normally changes orientation, such as from above the line to below the line. Rotations may also change orientation depending on the angle.**Linearity of enlargements and reductions**: Enlargements and reductions change the size of shapes. They maintain their same proportions and angles using a scale factor.

**Exploring transformative operations:**

**Combining transformations**: Several transformations to a shape can sometimes be simplified to a single change. For example, rather than making two reflections across two intersecting lines, we can just rotate the shape.**Sequential impact**: The order of transformations matters. Rotating a shape before a translation can lead to a different outcome than doing it the other way around.

**Practical example:**

### 7

If a parallelogram is rotated 90 degrees about its centre and then reflected over the x-axis, which properties are preserved?

## Transformations in maths in the real world

### Real-world applications

**Computer Graphics**: Transformations are at the heart of computer graphics. They allow objects to be rotated, scaled and positioned in digital space. In video game development, transformations are used to animate characters and manipulate objects within the game.**Architecture and Engineering**: Architects and engineers use transformations to see changes in scale and perspective. This means they can create models before construction begins to ensure the accuracy of measurements and the overall design.**Art and Design**: Some artists use transformations to create symmetrical designs and patterns. Reflections and rotations are particularly common in kaleidoscopic and tessellation art.

### Using transformations

**Mapping and Navigation**: GPS turns geographical data into different formats and perspectives using transformation. This creates more accurate map rendering and real-time positioning.**Medical Imaging**: Transformations are used in medical imaging to manipulate and enhance images from MRIs and CT scans. This gives more details views of the human body and allows for more accurate diagnoses.

**Practical example:**

### 8

How could transformation be used in planning a public park?

## Final thoughts on definitions of transformations

## Glossary

- Transformation - Any operation that moves or changes a shape while keeping certain properties the same.
- Reflection - A transformation that flips a figure over a line, creating a mirror image.
- Rotation - Turning a figure about a fixed point through a given angle and direction.
- Translation - A transformation that slides every point of a shape at a constant distance in a specified direction.
- Enlargement - A transformation that changes the size of a figure but not the shape. It's defined by a scale factor and a centre of enlargement.
- Scale Factor - A number which transform a figure, either through dilation or scaling it up. A scale factor greater than 1 enlarges the figure. A scale factor of less than 1 reduces it.
- Centre of Enlargement - The point in space from where points in a shape are bigger or smaller in an enlargement.
- Vector - A quantity that has direction and magnitude. It's used to describe the movement of points in a translation.
- Isometry - A transformation that keeps the distance between points so the original shape and the reflection are identical. Reflections, rotations and translations are all isometries.
- Congruent - Exactly equal in size and shape. Congruent figures are identical.
- Orientation - The arrangement of points in a specific direction inside a shape. Some transformations change the orientation of a figure e.g. reflection.
- Composite transformation - A sequence of two or more transformations performed one after the other.
- Inverse Transformation - A transformation that reverses the effect of a previous transformation, returning the shape to its original position or orientation.

*This post was updated on 05 Jul, 2024.*