Translation Transformation: Introduction and Point Translation
📌 Geometric transformation involves changes in position, shape, or size of geometric objects like points, lines, or areas.
📌 The four types of geometric transformations covered are translation (shifting), reflection (mirroring), rotation (turning), and dilation (scaling).
📌 Translation (T) shifts a point $(x, y)$ by a horizontal component $a$ and a vertical component $b$, resulting in a new point $(x', y')$ calculated as $x' = x + a$ and $y' = y + b$.
📌 To find the translated point $(x', y')$, add the initial coordinates $(x, y)$ to the translation vector $T = \begin{pmatrix} a \\ b \end{pmatrix}$: $(x', y') = (x, y) + (a, b)$.

Finding Original Coordinates and Composition of Translations
📌 If the translated point $(x', y')$ is known, the original point $(x, y)$ is found by subtracting the translation vector: $(x, y) = (x', y') - (a, b)$.
📌 Composition of translations involves adding multiple translation vectors sequentially; for $T_1$ followed by $T_2$, the final translation vector is $T_{total} = T_1 + T_2$.
📌 Example of composition: A point $P(4, 5)$ translated by $T_1 = \begin{pmatrix} 3 \\ -1 \end{pmatrix}$ and then by $T_2 = \begin{pmatrix} -2 \\ 2 \end{pmatrix}$ results in $P'' = (4+3-2, 5-1+2) = (5, 6)$.

Translation of Lines and Curves
📌 To translate a line equation (e.g., $Ax + By + C = 0$) by $T = \begin{pmatrix} a \\ b \end{pmatrix}$, first determine the inverse relationship for $x$ and $y$: $x = x' - a$ and $y = y' - b$.
📌 Substitute these expressions back into the original equation. For example, translating $2x + 3y - 6 = 0$ by $T = \begin{pmatrix} -2 \\ 3 \end{pmatrix}$ requires substituting $x = x' + 2$ and $y = y' - 3$.
📌 The resulting line equation after substitution and simplification is $2(x' + 2) + 3(y' - 3) - 6 = 0$, which simplifies to $2x' + 3y' - 11 = 0$ (dropping the prime notation yields $2x + 3y - 11 = 0$).

Key Points & Insights
➡️ Positive 'a' in $T(a, b)$ means shifting right (positive horizontal direction), while negative 'a' means shifting left.
➡️ Positive 'b' in $T(a, b)$ means shifting up (positive vertical direction), while negative 'b' means shifting down.
➡️ When finding the original point or line equation, reverse the sign of the translation vector components before subtracting.
➡️ For line translation, always invert the translation direction when substituting $x$ and $y$ into the original equation (if $T = \begin{pmatrix} a \\ b \end{pmatrix}$, substitute $x$ with $x-a$ and $y$ with $y-b$).

📸 Video summarized with SummaryTube.com on Nov 06, 2025, 04:45 UTC