Transformations, congruence, vectors and similarity Revision Notes

What is Reflection?

Reflection is a geometric transformation that flips a figure over a line, called the line of reflection. The reflected image is a mirror image of the original. The size and shape of the figure remain unchanged, but its orientation is reversed. Think of it like looking in a mirror: your reflection is the same size, but your left appears as your right.

Key Properties of Reflection

1. Distance: Every point on the original object is the same perpendicular distance from the line of reflection as its corresponding point on the image. 2. Shape and Size: The reflected figure is congruent to the original figure, meaning it has the exact same shape and size. 3. Orientation: The orientation of the figure is reversed. If the original figure was 'clockwise', the reflected figure will be 'anticlockwise'. This means the image cannot be rotated to match the original.

How to Perform a Reflection

To reflect a shape, follow these steps for each vertex (corner) of the shape: 1. Identify the line of reflection. 2. From a vertex, draw a perpendicular line to the line of reflection. 3. Measure the distance from the vertex to the line of reflection along this perpendicular line. 4. Measure the same distance again on the opposite side of the line of reflection to find the image point. 5. Repeat for all vertices. 6. Connect the image points to form the reflected shape. Use a ruler and pencil, and if on a grid, count squares to find the distances accurately.

Common Lines of Reflection

Understanding common lines of reflection helps in reflecting shapes efficiently: - x-axis (y=0): The y-coordinate of each point changes sign, e.g. (x, y) becomes (x, -y). - y-axis (x=0): The x-coordinate of each point changes sign, e.g. (x, y) becomes (-x, y). - Line y=x: The x and y coordinates are swapped, e.g. (x, y) becomes (y, x). - Line y=-x: The x and y coordinates are swapped and both signs are changed, e.g. (x, y) becomes (-y, -x). - Vertical lines (x=k) and Horizontal lines (y=k): You need to count the distance to the line and then count the same distance again on the other side.

What is Reflection?

Reflection is a transformation where every point of a shape is mapped to a point an equal distance on the opposite side of a 'mirror line'. The reflected image is a mirror image of the original. The key properties are that the size and shape of the object do not change, but its orientation is reversed. It's like flipping the shape over a line.

How to Reflect a Shape

To reflect a shape, you need a shape and a mirror line. For each vertex of the shape: 1. Measure the perpendicular distance from the vertex to the mirror line. 2. Measure the same distance from the mirror line on the opposite side, along the same perpendicular line. This point is the reflected vertex. Repeat for all vertices and connect them to form the reflected image. Using a ruler and protractor (or grid lines) helps ensure accuracy. For example, if reflecting over the x-axis, (x, y) becomes (x, -y). If reflecting over the y-axis, (x, y) becomes (-x, y).

Common Mirror Lines

You will often reflect shapes over specific lines. Examples include: the x-axis (where y-coordinates change sign), the y-axis (where x-coordinates change sign), the line y = x (where (x, y) becomes (y, x)), and horizontal or vertical lines like x = k or y = k. For a line like x = 3, if a point is at (1, 5), it is 2 units to the left of the mirror line. Its reflected point will be 2 units to the right, at (5, 5). Similarly for y = k.

Key Characteristics of Reflection

Reflections preserve the size, angles, and lengths of the segments of the original shape. This means the reflected image is congruent to the original. However, the 'handedness' or orientation of the shape is reversed. For instance, if you read text on a shape, in its reflection the text will appear reversed, suitable for reading in a mirror. Always ensure your reflected points are perpendicular to the mirror line and equidistant from it.

Elements of a Rotation

To completely describe any rotation, three pieces of information are essential: the centre of rotation, the angle of rotation, and the direction of rotation. The centre is the fixed point around which the shape turns. The angle is how many degrees the shape turns, for example, 90°, 180°, or 270°. The direction specifies whether the turn is clockwise (like a clock) or anti-clockwise (opposite to a clock). Without all three elements, a rotation cannot be uniquely identified or performed. Understanding these components is fundamental to applying rotation transformations.

Performing a Rotation

When performing a rotation, each point of the original figure (the object) moves to a new position to form the rotated figure (the image). The key is that every point on the object travels along a circular arc with the centre of rotation as its origin. The radius of this arc is the distance from the centre of rotation to the point. The length of the arc corresponds to the angle of rotation. For example, to rotate a point P 90° clockwise around a centre C, you draw a line from C to P. Then, maintaining the same distance CP, draw a new line from C at a 90° angle in the clockwise direction. The end of this new line is P'. Repeat this for all key vertices of the shape.

Identifying Rotations

To identify a rotation that has taken place between an object and its image, you need to find the centre, angle, and direction. To find the centre, you can join a point on the object to its corresponding point on the image. Then, draw the perpendicular bisector of this line segment. Repeat this for another pair of corresponding points. The intersection of these perpendicular bisectors is the centre of rotation. To find the angle and direction, you can connect the centre to any point on the object and its corresponding point on the image. The angle between these two lines, measured at the centre, is the angle of rotation, and its orientation indicates the direction (clockwise or anti-clockwise).

Using Tracing Paper for Rotations

Tracing paper is an invaluable tool for visualising and performing rotations. First, place the tracing paper over the original figure and the coordinate grid. Mark the centre of rotation clearly on the tracing paper. Trace the shape of the object onto the tracing paper. Next, firmly hold the tracing paper at the centre of rotation with a pen or pencil. Rotate the tracing paper in the specified direction (clockwise or anti-clockwise) by the given angle. Once rotated, press down the tracing paper firmly to transfer the rotated image onto the original paper, or carefully mark the new positions of the vertices.

Key Elements of a Rotation

To describe any rotation, you must identify three key elements: the centre of rotation, the angle of rotation, and the direction of rotation. The centre of rotation is the fixed point around which the shape turns; it can be any point on the coordinate grid or even within the shape itself. The angle specifies how much the shape turns, usually given in degrees (e.g., 90°, 180°, 270°). The direction indicates whether it's a clockwise (CW) or anti-clockwise (ACW) turn.

Performing Rotations on a Coordinate Grid

When rotating a shape on a coordinate grid, it's crucial to first locate the centre of rotation. Then, for each vertex of the shape, imagine a line connecting it to the centre of rotation. Rotate this line by the given angle and direction. The new position of the vertex will be along this rotated line, maintaining the same distance from the centre of rotation. Using tracing paper is a highly effective method: place it over the original shape and axes, mark the centre of rotation, trace the shape, and then physically turn the tracing paper around the marked centre.

Common Rotation Angles and Their Effects

Understanding common rotation angles simplifies the process. A 90° clockwise rotation is equivalent to a 270° anti-clockwise rotation and vice versa. For example, if a point (x, y) is rotated 90° clockwise about the origin (0,0), its new coordinates become (y, -x). If rotated 90° anti-clockwise, it becomes (-y, x). A 180° rotation about the origin changes (x, y) to (-x, -y); this rotation is identical whether clockwise or anti-clockwise. Familiarity with these specific transformations, especially from the origin, can speed up calculations.