Geometry and Measure - Position and movement (Transformations)

A triangle with vertices $A(1, 2)$, $B(3, 2)$, and $C(1, 4)$ is translated by the vector $\begin{pmatrix} 2 \\ -3 \end{pmatrix}$. Find the new coordinates of $A'$.

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

To translate a point, add the $x$ component of the vector to the $x$-coordinate and the $y$ component of the vector to the $y$-coordinate.

Reflect the point $P(4, 5)$ in the line $y = x$.

$P'(5, 4)$

When reflecting in the line $y = x$, the $x$ and $y$ coordinates are swapped.

A square has a side length of $5$ cm. It is enlarged by a scale factor of $3$. What is the side length of the new square?

$5 \text{ cm} \times 3 = 15 \text{ cm}$

To find the new length after enlargement, multiply the original length by the scale factor $k$.

Rotate the point $(2, 0)$ $90^\circ$ anti-clockwise about the origin $(0,0)$.

$(0, 2)$

A $90^\circ$ anti-clockwise rotation moves a point from the positive $x$-axis to the positive $y$-axis.

1. Identify the distance of each point from the line $y = 3$.
2. Point $A(1, 1)$ is $2$ units below the line, so $A'$ will be $2$ units above the line at $(1, 3+2) = (1, 5)$.
3. Point $B(3, 1)$ is $2$ units below the line, so $B'$ will be $(3, 5)$.
4. Point $C(3, 2)$ is $1$ unit below the line, so $C'$ will be $(3, 4)$.
5. Point $D(1, 2)$ is $1$ unit below the line, so $D'$ will be $(1, 4)$. Final coordinates: $A'(1, 5), B'(3, 5), C'(3, 4), D'(1, 4)$.

Reflecting across a horizontal line $y=k$ keeps the $x$-coordinate the same while the $y$-coordinate changes such that the line is the midpoint.

For a rotation of $180^\circ$ about the origin, the rule $(x, y) \rightarrow (-x, -y)$ applies.

1. $X(1, 1) \rightarrow X'(-1, -1)$
2. $Y(1, 3) \rightarrow Y'(-1, -3)$
3. $Z(2, 1) \rightarrow Z'(-2, -1)$

A $180^\circ$ rotation is equivalent to reflecting in both the $x$ and $y$ axes sequentially, effectively negating both coordinates.