Transformations, congruence, vectors and similarity Revision Notes

What is a Translation?

A translation is a geometric transformation that moves every point of a figure or a shape by the same distance in a given direction. It's like sliding an object without rotating, resizing, or flipping it. The original shape and the translated image will be congruent, meaning they have the same size and shape, and face the same direction. The position is the only aspect that changes during a translation.

Describing Translations with Vectors

Translations are described using a translation vector, which is often written in column form like $\begin{pmatrix} x \\ y \end{pmatrix}$. The top number, 'x', tells you the horizontal movement: positive for right, negative for left. The bottom number, 'y', tells you the vertical movement: positive for up, negative for down. For example, $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ means move 3 units right and 2 units down.

Translating Points and Shapes

To translate a single point with coordinates $(a, b)$ by a vector $\begin{pmatrix} x \\ y \end{pmatrix}$, you simply add the components of the vector to the coordinates of the point. The new coordinates will be $(a+x, b+y)$. To translate an entire shape, you should apply the translation vector to each vertex (corner point) of the shape. Once all the new vertex coordinates are found, connect them in the same order as the original shape to draw the translated image.

Key Properties of Translations

When a shape is translated, its orientation remains the same. This means it doesn't spin or flip. The size and shape of the object also stay identical. Translations are examples of 'isometries', transformations that preserve distance and angle measures. This ensures that the original figure and its translated image are congruent.

Order Matters for Combined Transformations

When working with multiple transformations, the order of application is critical. Applying transformation A then transformation B will almost always yield a different final result than applying B then A. For example, reflecting a shape in the x-axis and then rotating it 90 degrees clockwise about the origin produces a different image than rotating it first and then reflecting it. Always follow the specified order carefully, applying one transformation at a time to the *current* image.

Step-by-Step Application

To successfully find the image of a shape after a sequence of transformations, it's best to work step-by-step. First, apply the initial transformation to the original object to find the first image. Then, use this first image as the object for the second transformation to find the final image. If there are more transformations, continue this process sequentially. Drawing intermediate images (even lightly) can help prevent errors and visualize the process clearly.

Identifying Single Equivalent Transformations

Sometimes, a combination of two transformations can be equivalent to a single, simpler transformation. For example: two sequential reflections in parallel lines are equivalent to a single translation. The translation vector will be perpendicular to the reflection lines and have a magnitude twice the distance between the lines. Two sequential reflections in intersecting lines are equivalent to a single rotation. The centre of rotation is the intersection point of the lines, and the angle of rotation is twice the angle between the lines.

Describing Combined Transformations

When asked to describe a sequence of transformations, you must fully describe each individual transformation and state the order in which they occur. For a translation, give the vector. For a reflection, give the equation of the mirror line. For a rotation, give the centre, angle, and direction. For an enlargement, give the centre and scale factor. Provide all necessary information clearly for each step.

What 'Congruent' Means

Two geometric figures are congruent if they have exactly the same size and shape. For triangles, this means that if triangle A is congruent to triangle B, then reversing, rotating, or translating triangle A could make it perfectly overlap triangle B. All corresponding sides must be equal in length, and all corresponding angles must be equal in measure. However, you don't need to check all six conditions (3 sides, 3 angles) every time. Instead, there are specific minimum conditions, called congruence criteria, that allow you to prove congruence.

Criteria for Congruent Triangles

There are four main congruence criteria used to determine if two triangles are congruent without checking all corresponding parts:

1. SSS (Side-Side-Side): If all three corresponding sides of two triangles are equal in length.
2. SAS (Side-Angle-Side): If two corresponding sides and the 'included' angle (the angle between those two sides) are equal. The order matters here.
3. ASA (Angle-Side-Angle): If two corresponding angles and the 'included' side (the side between those two angles) are equal. The order also matters here.
4. RHS (Right-angle-Hypotenuse-Side): Specifically for right-angled triangles; if the hypotenuse and one other corresponding side are equal in length. The right angle is the 'R' part.

Applying Congruence in Problem Solving

When faced with a problem involving triangles, look for clues that suggest congruence. Identify known side lengths and angle measures. Compare these between the two triangles in question. Then, try to match them against one of the four congruence criteria (SSS, SAS, ASA, RHS). Clearly state which criterion you are using as justification. Once you've proven two triangles are congruent, you can use this fact to deduce the values of any unknown corresponding sides or angles in those triangles, as they must be equal. Remember to always state your reasons clearly.

Common Pitfalls

Be careful not to confuse congruence with similarity. Similar triangles have the same shape but different sizes (corresponding angles are equal, but sides are in proportion). Also, make sure the angle in SAS and the side in ASA are 'included'. For SAS, the angle must be between the two given sides. For ASA, the side must be between the two given angles. SSA (Side-Side-Angle) is generally NOT a valid congruence criterion, as it can lead to two possible triangle constructions, unless it's the special case of RHS for right-angled triangles.

What 'Congruent' Means

Congruent triangles are triangles that are identical in both shape and size. This means that if you could pick one up and place it on top of the other, they would match up perfectly. All corresponding sides must be equal in length, and all corresponding angles must be equal in measure. It's important not to confuse congruence with similarity; similar triangles have the same shape but different sizes.

The Four Congruence Rules

To prove two triangles are congruent, you don't need to show that all three sides and all three angles are equal. Instead, you can use one of these four rules:

1. SSS (Side-Side-Side): If all three sides of one triangle are equal to the three corresponding sides of another triangle.
2. SAS (Side-Angle-Side): If two sides and the *included* angle (the angle between those two sides) of one triangle are equal to the two corresponding sides and included angle of another triangle.
3. ASA (Angle-Side-Angle): If two angles and the *included* side (the side between those two angles) of one triangle are equal to two corresponding angles and the included side of another triangle.
4. AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle are equal to two corresponding angles and the non-included side of another triangle. (Note: AAS is sometimes combined with ASA, as knowing two angles implies the third is also equal). It is crucial to correctly identify corresponding parts.

Applying Congruence Rules

When attempting to prove triangles are congruent, first identify the triangles you are comparing. Look for given information about side lengths or angle measures. Also, look for properties like common sides, vertically opposite angles, or alternate/corresponding angles formed by parallel lines that might provide additional equal parts. Clearly state which rule you are using (SSS, SAS, ASA, or AAS) and list the three corresponding equalities you have identified to satisfy that rule. For example, to prove by SAS, you'd state Side 1 = Side 1, Included Angle = Included Angle, Side 2 = Side 2.