Creating highly detailed study notes with a focus on mathematical equations and step-by-step solutions for the Translation topic in the CIE IGCSE syllabus, particularly with a preference for solutions presented in equation form, is a unique challenge. Here, I'll provide a structured approach to explaining translations through mathematical examples, emphasizing equations and calculations. The objective is to create a compact yet comprehensive guide for students, focusing on the practical application of translation vectors to geometric figures.

Introduction to Translation

Translation is a type of geometric transformation that moves every point of a shape a consistent distance in a specified direction. The shape's size and orientation remain unchanged.

Image courtesy of Unacademy

Understanding Translation

Translation is defined by a vector $\vec{v} = (x, y)$, which specifies the movement along the horizontal (x-axis) and vertical (y-axis) directions.

• Notation: A vector $(x, y)$ describes how far to move along the x-axis and y-axis.
• Point Translation: For point $P(a, b)$, a translation by $(x, y)$ results in point $P'(a+x, b+y)$.

Examples of Translation

Example 1: Translating a Point

Given:

• Point $A(2, 3)$
• Translation vector $(3, -2)$

Objective: Find $A'$ after translation.

Solution steps:

1. Original coordinates of $A$: $A = (2, 3)$

2. Translation vector: $\vec{v} = (3, -2)$

3. Apply translation: $A' = A + \vec{v}$

4. Calculate: $A' = (2 + 3, 3 - 2) = (5, 1)$

Example 2: Translating a Shape

Given:

• Triangle vertices: $A(1, 2)$, $B(4, 5)$, $C(7, 2)$
• Translation vector $(2, -3)$

Objective: Find new positions of $A'$, $B'$, and $C'$.

Solution steps:

1. Translate $A$: $A' = (1 + 2, 2 - 3) = (3, -1)$

2. Translate $B$: $B' = (4 + 2, 5 - 3) = (6, 2)$

3. Translate $C$: $C' = (7 + 2, 2 - 3) = (9, -1)$

Practice Problems

1. Translate point $D(4, -5)$ with vector $(-3, 6)$.

$D(4, -5)$ with vector $(-3, 6)$

$D' = D + \vec{v} = (4 - 3, -5 + 6) = (1, 1)$

2. Rectangle vertices $E(1, 1)$, $F(1, 4)$, $G(5, 4)$, $H(5, 1)$ with vector $(2, -2)$. Find new vertices.

1. $E' = (1 + 2, 1 - 2) = (3, -1)$

2. $F' = (1 + 2, 4 - 2) = (3, 2)$

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

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