Transformations, congruence, vectors and similarity Revision Notes

What is Congruence?

Congruence in geometry means that two figures are identical in shape and size. For triangles, this means all corresponding sides are equal in length, and all corresponding angles are equal in measure. If two triangles are congruent, you can superimpose one perfectly onto the other.

Four Congruence Conditions (SSS, SAS, ASA, RHS)

There are four main criteria to prove that two triangles are congruent without needing to check every side and angle:

1. SSS (Side-Side-Side): If all three corresponding sides are equal.
2. SAS (Side-Angle-Side): If two corresponding sides and the included angle (the angle between those two sides) are equal.
3. ASA (Angle-Side-Angle): If two corresponding angles and the included side (the side between those two angles) are equal.
4. RHS (Right-angle-Hypotenuse-Side): If both triangles are right-angled, and their hypotenuses and one other pair of corresponding sides are equal.

Applying Congruence for Problem Solving

Once you've proven two triangles are congruent using one of the four conditions, you can deduce that all their other corresponding parts (sides and angles) are also equal. This is often used to find unknown lengths or angles in complex diagrams. Look for common sides, vertically opposite angles, parallel lines forming alternate or corresponding angles, and shared vertices to identify corresponding parts.

Common Pitfalls

Be careful not to confuse ASA with AAS. While AAS is also a valid congruence condition (Angle-Angle-Side, where the side is not included between the two angles), it can be derived from ASA because if two angles are known, the third angle is also known. Also, SSA (Side-Side-Angle) is NOT a valid congruence condition, as it can result in two different triangles (ambiguous case).

What is a Vector?

A vector is a quantity that has both magnitude (size or length) and direction. Unlike scalars (which only have magnitude, e.g., speed, temperature), vectors specify 'how much' and 'in what way'. Vectors can be represented graphically by an arrow, where the length of the arrow represents the magnitude and the arrowhead indicates the direction. In coordinate form, a 2D vector is often written as a column vector, such as $\begin{pmatrix} x \\ y \end{pmatrix}$, where 'x' is the horizontal displacement and 'y' is the vertical displacement. For example, the vector $\begin{pmatrix} 4 \\ 1 \end{pmatrix}$ represents a movement of 4 units to the right and 1 unit up.

Vector Addition and Subtraction

To add two vectors, you add their corresponding components. If you have vector $\mathbf{a} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix}$ and vector $\mathbf{b} = \begin{pmatrix} x_2 \\ y_2 \end{pmatrix}$, then their sum is $\mathbf{a} + \mathbf{b} = \begin{pmatrix} x_1 + x_2 \\ y_1 + y_2 \end{pmatrix}$. Geometrically, imagine placing the tail of vector $\mathbf{b}$ at the head of vector $\mathbf{a}$; the resultant vector $\mathbf{a} + \mathbf{b}$ goes from the tail of $\mathbf{a}$ to the head of $\mathbf{b}$. For subtraction, you subtract their corresponding components: $\mathbf{a} - \mathbf{b} = \begin{pmatrix} x_1 - x_2 \\ y_1 - y_2 \end{pmatrix}$. Alternatively, subtracting a vector is the same as adding its negative, i.e., $\mathbf{a} - \mathbf{b} = \mathbf{a} + (-\mathbf{b})$, where $-\mathbf{b} = \begin{pmatrix} -x_2 \\ -y_2 \end{pmatrix}$.

Scalar Multiplication of Vectors

Multiplying a vector by a scalar (a real number) changes its magnitude but usually not its direction. If the scalar is positive, the direction remains the same. If the scalar is negative, the direction of the vector is reversed. For a scalar $k$ and a vector $\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}$, the product is $k\mathbf{v} = \begin{pmatrix} kx \\ ky \end{pmatrix}$. For example, $3 \begin{pmatrix} 2 \\ 5 \end{pmatrix} = \begin{pmatrix} 3 \times 2 \\ 3 \times 5 \end{pmatrix} = \begin{pmatrix} 6 \\ 15 \end{pmatrix}$. This new vector is three times longer than the original and points in the same direction. If the scalar is -2, then $-2 \begin{pmatrix} 2 \\ 5 \end{pmatrix} = \begin{pmatrix} -4 \\ -10 \end{pmatrix}$, which is twice as long and points in the opposite direction.

Core method

This is Higher-tier content, so solutions should communicate multi-step reasoning precisely. Vectors describe displacement with magnitude and direction. Add vectors by combining components or following consecutive journeys, subtract by adding the opposite vector, and multiply by scalars component by component. In geometric proofs, express routes between points in terms of a small set of base vectors. The essential approach is to identify the mathematical structure before calculating, retain exact values where possible, and show enough working for each conclusion to be checked independently.

Worked reasoning and checks

If a = (3, −2) and b = (−1, 5), then a + b = (2, 3), 2a = (6, −4), and a − b = (4, −7). Parallel vectors are scalar multiples. Points are collinear when the vectors along the same line are scalar multiples with a common point. Check the final result against the original information, including signs, units, scale and whether the requested form or degree of accuracy has been supplied.

What is a Column Vector?

A column vector is a mathematical tool used to represent a translation or displacement. It has two components: a horizontal component (top number) and a vertical component (bottom number). It's written in the form $\begin{pmatrix} x \\ y \end{pmatrix}$. The $x$ value tells you how far to move horizontally (positive for right, negative for left), and the $y$ value tells you how far to move vertically (positive for up, negative for down). Each component is an integer for foundation level. For example, $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ means move 3 units right and 2 units down.

Adding and Subtracting Column Vectors

To add two column vectors, you simply add their corresponding components. This means adding the top numbers together and adding the bottom numbers together. For example, if $\mathbf{a} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ -5 \end{pmatrix}$, then $\mathbf{a} + \mathbf{b} = \begin{pmatrix} 2+1 \\ 3+(-5) \end{pmatrix} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}$.

Similarly, to subtract two column vectors, you subtract their corresponding components. So, $\mathbf{a} - \mathbf{b} = \begin{pmatrix} 2-1 \\ 3-(-5) \end{pmatrix} = \begin{pmatrix} 1 \\ 8 \end{pmatrix}$. Thinking of subtraction as adding the negative of a vector can also be helpful: $\mathbf{a} - \mathbf{b} = \mathbf{a} + (-\mathbf{b})$, where $-\mathbf{b} = \begin{pmatrix} -1 \\ 5 \end{pmatrix}$.

Multiplying a Vector by a Scalar

When you multiply a column vector by a scalar (a single number), you multiply each component of the vector by that scalar. This scales the length of the vector and can reverse its direction. For instance, if $\mathbf{v} = \begin{pmatrix} 4 \\ -1 \end{pmatrix}$ and you multiply it by the scalar $3$, then $3\mathbf{v} = 3 \begin{pmatrix} 4 \\ -1 \end{pmatrix} = \begin{pmatrix} 3 \times 4 \\ 3 \times -1 \end{pmatrix} = \begin{pmatrix} 12 \\ -3 \end{pmatrix}$.

If the scalar is negative, the direction of the vector is reversed. For example, $-2\mathbf{v} = -2 \begin{pmatrix} 4 \\ -1 \end{pmatrix} = \begin{pmatrix} -2 \times 4 \\ -2 \times -1 \end{pmatrix} = \begin{pmatrix} -8 \\ 2 \end{pmatrix}$. This vector is twice as long as $\mathbf{v}$ and points in the opposite direction.