Translations are transformations that move points or objects in a plane without changing their shape or orientation. This section focuses on understanding translations through vector notation, crucial for IGCSE students mastering geometry.

Introduction to Translations

In geometry, a translation moves a figure or point from one location to another on the plane. This movement is described using vectors, providing a clear mathematical representation of the translation process.

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Understanding Vector Notation

Vector Basics

• A vector represents both direction and magnitude.
• In 2D, vectors are written as $(x, y)$, where $x$ and $y$ indicate horizontal and vertical movements.

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Vectors and Translations

• A translation vector $\vec{AB} = (x, y)$ shows the shift from point $A$ to point $B$.
• $x$ and $y$ detail the movement along the x-axis and y-axis.

Applying Translations

Translation Formula

For any point $P(x_1, y_1)$ translated by $\vec{v} = (x, y)$, the new position $P'$ is calculated as: $P' = (x_1 + x, y_1 + y)$

Worked Examples

Translating a Point

Given: Point $A(2, 3)$, Vector $\vec{v} = (4, -2)$

Find: New position $A'$

Solution:

$A' = (2 + 4, 3 - 2) = (6, 1)$

Hence, $A$ moves to$A'(6, 1)$.

Translating a Shape

Given: Triangle vertices $A(1, 2)$, $B(3, 5)$, $C(5, 3)$, Vector$\vec{v} = (-2, 3)$

Find: New vertices positions

Solution:

• $A'$: $A' = (1 - 2, 2 + 3) = (-1, 5)$
• $B'$: $B' = (3 - 2, 5 + 3) = (1, 8)$
• $C'$: $C' = (5 - 2, 3 + 3) = (3, 6)$

New vertices: $A'(-1, 5)$, $B'(1, 8)$, $C'(3, 6)$.

Practice Problems

Problem 1

Given: $D(4, -1)$, $\vec{v} = (-3, 4)$

Find: $D'$

Solution:

$D' = (4 - 3, -1 + 4) = (1, 3)$

Problem 2

Given: Square vertices$P(1, 1)$, $Q(4, 1)$, $R(4, 4)$, $S(1, 4)$, $\vec{v} = (2, -3)$

Find: New vertices positions

Solution:

• $P'$: $P' = (1 + 2, 1 - 3) = (3, -2)$
• $Q'$: $Q' = (4 + 2, 1 - 3) = (6, -2)$
• $R'$: $R' = (4 + 2, 4 - 3) = (6, 1)$
• $S'$: $S' = (1 + 2, 4 - 3) = (3, 1)$

New square vertices: $P'(3, -2)$, $Q'(6, -2)$, $R'(6, 1)$, $S'(3, 1)$.

Success Tips

• Draw diagrams to visualize translations.
• Solve various problems for practice.
• Ensure understanding of vector operations (addition, scalar multiplication).