Transformations: Question 1

Worked solution

For each part, first check the image is congruent to $T$ (a right-angled triangle with legs $3$ and $2$), then identify the one transformation and give every required detail.

Triangle $T$: right angle at $(2,1)$, vertical leg $(2,1)\to(2,4)$ of length $3$, horizontal leg $(2,1)\to(4,1)$ of length $2$.

(a) Triangle $U$: $(5,-4),(5,-1),(7,-4)$

The orientation and shape are unchanged, so this is a translation. The translation vector is found by subtracting an object point from its image point:

$\begin{pmatrix} 5 \\ -4 \end{pmatrix}-\begin{pmatrix} 2 \\ 1 \end{pmatrix}=\begin{pmatrix} 3 \\ -5 \end{pmatrix}$

Check with another pair of vertices:

$\begin{pmatrix} 7 \\ -4 \end{pmatrix}-\begin{pmatrix} 4 \\ 1 \end{pmatrix}=\begin{pmatrix} 3 \\ -5 \end{pmatrix}\ \checkmark$

Description: a translation by the column vector $\boxed{\begin{pmatrix} 3 \\ -5 \end{pmatrix}}$

(b) Triangle $V$: $(1,2),(4,2),(1,4)$

The orientation is reversed (the right angle moves from $(2,1)$ to $(1,2)$), so this is a reflection. Match the vertices:

$(2,1)\to(1,2),\qquad (2,4)\to(4,2),\qquad (4,1)\to(1,4)$

Each image swaps the $x$- and $y$-coordinates of its object point: $(x,y)\to(y,x)$. This is reflection in the line $y=x$. Check the midpoint of $(2,1)$ and $(1,2)$ is $(1.5,1.5)$, which lies on $y=x$. ✓

Description: a reflection in the line $\boxed{y=x}$

(c) Triangle $W$: $(-1,2),(-4,2),(-1,4)$

The shape is congruent and the orientation is preserved (not a reflection), but it is not a translation, so this is a rotation. Match the vertices:

$(2,1)\to(-1,2),\qquad (2,4)\to(-4,2),\qquad (4,1)\to(-1,4)$

Each image follows the rule $(x,y)\to(-y,x)$, which is a quarter turn anticlockwise about the origin. For example $(2,1)$ in the first quadrant maps to $(-1,2)$ in the second quadrant, confirming the anticlockwise direction. The centre $(0,0)$ is equidistant from each object–image pair, e.g. $(2,1)$ and $(-1,2)$ are both a distance $\sqrt{5}$ from the origin. ✓

A rotation must state the angle, the direction and the centre.

Description: a rotation through $90^\circ$ anticlockwise, centre $\boxed{(0,0)}$