Transformations
🎯 In this topic you will
- Transform shapes using combinations of reflections, translations, and rotations
- Identify and describe different types of transformations
- Explain that after a combination of transformations the object and its image remain congruent
🔄 Describing Transformations
You already know how to describe the transformation that maps an object on to its image. Here is a quick reminder:
| To describe | You must give |
|---|---|
| a reflection | the equation of the mirror line |
| a translation | the column vector |
| a rotation |
|
You can use a combination of reflections, translations and rotations to transform a shape.
❓ EXERCISES
1. The diagram shows triangle A. Cards a, b and c show three combinations of transformations. Diagrams i, ii and iii show triangle A and its image, triangle B. Match each card with the correct diagram.
a. A translation $\begin{bmatrix}2 \\ -1\end{bmatrix}$ followed by a reflection in the line $y=3$.
b. A rotation of $90^\circ$ anticlockwise, centre $(3,5)$ followed by a translation $\begin{bmatrix}3 \\ -2\end{bmatrix}$.
c. A reflection in the line $x=4$ followed by a rotation of $180^\circ$, centre $(6,4)$.
👀 Show answer
a. Matches diagram ii.
b. Matches diagram iii.
c. Matches diagram i.
❓ EXERCISES
2. The diagram shows shape A on a coordinate grid.
Make two copies of the diagram.
On different copies of the diagram, draw the image of A after each combination of transformations.
a. Reflection in the $y$-axis followed by the translation $\begin{bmatrix}1 \\ -2\end{bmatrix}$.
b. Rotation of $90^\circ$ anticlockwise, centre $(-1,2)$, followed by reflection in the line $x=1$.
c. Look at your answers to parts a and b. In each part, are the object and the image congruent or not congruent? Explain your answers.
👀 Show answer
a. Reflect A in the $y$-axis to get A′. Then translate A′ by the vector $\begin{bmatrix}1 \\ -2\end{bmatrix}$ (right $1$, down $2$) to obtain the image.
b. Rotate A by $90^\circ$ anticlockwise about $(-1,2)$ to get A′. Then reflect A′ in the line $x=1$ to obtain the image.
c. In both parts the transformations are compositions of isometries (reflections/translations/rotations), so lengths and angles are preserved; therefore the object and image are congruent.
3. The diagram shows triangle B on a coordinate grid.
Make four copies of the diagram.
On different copies of the diagram, draw the image of B after each combination of transformations.
a. Translation $\begin{bmatrix}5 \\ 1\end{bmatrix}$, followed by reflection in the $x$-axis.
b. Rotation of $180^\circ$, centre $(-3,-2)$, followed by reflection in the $x$-axis.
c. Translation $\begin{bmatrix}-1 \\ 5\end{bmatrix}$ followed by rotation $90^\circ$ clockwise, centre $(-2,1)$.
d. Reflection in the line $y=-1$, followed by rotation $90^\circ$ anticlockwise, centre $(2,2)$.
👀 Show answer
a. Translate B by $\begin{bmatrix}5 \\ 1\end{bmatrix}$ (right $5$, up $1$), then reflect the image in the $x$-axis.
b. Rotate B by $180^\circ$ about $(-3,-2)$, then reflect the image in the $x$-axis.
c. Translate B by $\begin{bmatrix}-1 \\ 5\end{bmatrix}$, then rotate the image $90^\circ$ clockwise about $(-2,1)$.
d. Reflect B in the line $y=-1$, then rotate the image $90^\circ$ anticlockwise about $(2,2)$.
Each sequence is an isometry (or a composition of isometries), so the final image of B is congruent to B.
🧠 Think like a Mathematician
Task: The diagram shows three shapes, $X$, $Y$ and $Z$, on a coordinate grid. Make three clean copies of the grid. On the first grid draw shape $X$, on the second grid draw shape $Y$ and on the third grid draw shape $Z$.
Part a. On the first grid, draw the image of $X$ after the combination of transformations
Part b. On the second grid, draw the image of $Y$ after the combination of transformations
Part c. On the third grid, draw the image of $Z$ after the combination of transformations
Part d.
👀 Show Answer
- a.i / a.ii: The two final images are different. A reflection in $y=1$ and a rotation $90^\circ$ about $(2,-3)$ do not commute, so reversing the order changes the result.
- b.i / b.ii: The two final images are different. A reflection in $x=-1$ and a translation by $\begin{bmatrix}2\\-5\end{bmatrix}$ generally do not commute.
- c.i / c.ii: The two final images are different because the rotation $180^\circ$ about $(0,0)$ and reflection in $y=2$ do not share a common centre/axis; order matters.
- d.i: In each of parts $a$, $b$ and $c$, the results of $i$ and $ii$ are different.
- d.ii: Yes — the order can matter when combining transformations (e.g., reflection with translation, or rotation with reflection). Only in special cases do two different transformations give the same result in either order.
- d.iii: The sketches confirm non-commutativity: performing transformations in reverse order usually produces a different image.
- d.iv (one valid pair that works in any order): A rotation of $180^\circ$ about $(0,0)$ and a reflection in any line through $(0,0)$ (e.g., the $x$-axis $y=0$). These commute, so doing them in either order gives the same final image.
- d.v: When you apply the two transformations in both orders to $Z$, the final positions coincide — this verifies the claim in d.iv.
❓ EXERCISES
5. The diagram shows shapes $G$, $H$, $I$, $J$ and $K$ on a coordinate grid. Describe the transformation that transforms
a. shape $G$ to shape $H$
b. shape $G$ to shape $K$
c. shape $H$ to shape $J$
d. shape $J$ to shape $I$
Tip
Remember, for a reflection you need to write “Reflection” and give the equation of the mirror line.👀 Show answer
a. Reflection in the $y$-axis.
b. Translation by column vector $\begin{bmatrix}\Delta x\\ \Delta y\end{bmatrix}$, found by counting horizontal then vertical steps from $G$ to $K$ on the grid.
c. Translation by column vector $\begin{bmatrix}\Delta x\\ \Delta y\end{bmatrix}$ from $H$ to $J$.
d. Either a translation (give its vector) or—if distances to a fixed line are equal—a reflection in that line (state its equation), determined from the grid. Record the exact values from your diagram.
6. The diagram shows shapes $L$, $M$, $N$, $P$ and $Q$ on a coordinate grid. Describe the transformation that transforms
a. shape $L$ to shape $M$
b. shape $M$ to shape $N$
c. shape $N$ to shape $P$
d. shape $P$ to shape $Q$
e. shape $Q$ to shape $M$
f. shape $P$ to shape $M$
Tip
Remember, for a translation you need to write “Translation” and give the column vector.👀 Show answer
b. Translation $\begin{bmatrix}\Delta x\\ \Delta y\end{bmatrix}$ from $M$ to $N$.
c. Translation $\begin{bmatrix}\Delta x\\ \Delta y\end{bmatrix}$ from $N$ to $P$.
d. Translation $\begin{bmatrix}\Delta x\\ \Delta y\end{bmatrix}$ from $P$ to $Q$.
e. Translation $\begin{bmatrix}\Delta x\\ \Delta y\end{bmatrix}$ from $Q$ to $M$.
f. Translation $\begin{bmatrix}\Delta x\\ \Delta y\end{bmatrix}$ from $P$ to $M$.
7. The diagram shows triangles $R$, $S$, $T$, $U$, $V$ and $W$ on a coordinate grid. Describe the transformation that transforms
a. triangle $S$ to triangle $R$
b. triangle $R$ to triangle $T$
c. triangle $R$ to triangle $U$
d. triangle $V$ to triangle $T$
e. triangle $V$ to triangle $W$
Tip
Remember, for a rotation you need to write “Rotation” and give the centre of rotation as well as the angle of rotation and the direction of rotation.
👀 Show answer
a. Rotation about a stated centre by angle and direction that carries $S$ onto $R$ (write: Rotation, centre $(h,k)$, angle $\theta^\circ$, direction).
b. Translation/reflection/rotation as appropriate from $R$ to $T$; specify vector or line/centre and angle after inspecting the grid.
c. Describe the single transformation mapping $R$ to $U$ (state all required details).
d. Transformation that maps $V$ to $T$ (give full details).
e. Transformation that maps $V$ to $W$ (give full details).
Note: All listed transformations are isometries; the images are congruent to their originals.🧠 Think like a Mathematician
Task: The diagram shows shapes A, B, C, D and E on a coordinate grid.
a. Describe the single transformation that transforms
b. Reflect on your answers to part a. Is it possible to have more than one valid description of each transformation? Why or why not?
c. Describe a combined transformation that transforms
d. Reflect on your answers to part c. Did you find multiple possible combined transformations? Why might that be the case?
👀 Show Answers
- a.i: Shape A to E is a translation left and down. From the grid, the vector is $\begin{bmatrix}-4\\-4\end{bmatrix}$.
- a.ii: Shape B to C is a translation right and down. Vector: $\begin{bmatrix}2\\-4\end{bmatrix}$.
- a.iii: Shape C to D is a translation right by 2. Vector: $\begin{bmatrix}2\\0\end{bmatrix}$.
- b: Transformations such as translations are unique in their vector, but reflections or rotations can sometimes give the same result in different ways. This is why more than one description may be possible.
- c.i: Shape B to D can be described as the composition of the translation $\begin{bmatrix}2\\-4\end{bmatrix}$ (B→C) followed by $\begin{bmatrix}2\\0\end{bmatrix}$ (C→D). Combined vector: $\begin{bmatrix}4\\-4\end{bmatrix}$.
- c.ii: Shape B to E can be described as translation $\begin{bmatrix}-4\\-4\end{bmatrix}$ (A→E) composed with B→A move. Equivalent vector: $\begin{bmatrix}-8\\-4\end{bmatrix}$.
- c.iii: Shape C to A is a translation up and left. Vector: $\begin{bmatrix}-6\\4\end{bmatrix}$.
- d: Yes — for combined moves, you could describe them step by step (e.g., via intermediate shapes) or simplify into a single translation. Both are valid, so there can be more than one correct answer.
❓ EXERCISES
9. The vertices of triangle $G$ are at $(1,3)$, $(3,3)$ and $(1,4)$. The vertices of triangle $H$ are at $(-1,1)$, $(-3,1)$ and $(-1,2)$.
Sofia says: “I think the combined transformation to take $G$ to $H$ is: reflection in the line $x=-1$ then translation $\begin{bmatrix}2\\-2\end{bmatrix}$.”
Zara says: “I think the combined transformation to take $G$ to $H$ is: translation $\begin{bmatrix}0\\-2\end{bmatrix}$ then reflection in the $y$-axis.”
a. Who is correct? Explain your answer.
b. Write two different combined transformations that take $G$ to $H$.
c. How many more different combined transformations are there that take $G$ to $H$? Explain your answer.
👀 Show answer
a.Both are correct. Check a vertex, e.g. $(1,3)$. Sofia: reflect in $x=-1$ gives $(-3,3)$, then translate $\begin{bmatrix}2\\-2\end{bmatrix}$ gives $(-1,1)$. Zara: translate to $(1,1)$, then reflect in the $y$-axis gives $(-1,1)$. The other vertices match similarly.
b. Two valid combined transformations (any order stated): • Reflection in $x=-1$ then translation $\begin{bmatrix}2\\-2\end{bmatrix}$. • Translation $\begin{bmatrix}0\\-2\end{bmatrix}$ then reflection in the $y$-axis.
c.Infinitely many. You can prepend/append any isometry and its inverse (e.g., a rotation by $\theta$ followed by rotation by $-\theta$), or split the translation into parts, creating new two-step combinations that still map $G$ to $H$.
10. The diagram shows shape $A$ on a coordinate grid. Make two copies of the grid.
a. i. On the first copy, reflect shape $A$ in the line $y=5$ and label the image $B$. Then rotate shape $B$$180^\circ$, centre $(5,5)$, and label the image $C$.
a. ii. Describe the single transformation that takes shape $A$ to shape $C$.
b. i. On the second copy, rotate shape $A$$90^\circ$ anticlockwise, centre $(2,8)$, and label the image $D$. Then translate shape $D$$\begin{bmatrix}-3\\-3\end{bmatrix}$ and label the image $E$.
b. ii. Describe the single transformation that takes shape $A$ to shape $E$.
👀 Show answer
a. ii. A single reflection in the line$x=5$. Reason: reflect in $y=5$ sends $(x,y)\mapsto(x,\,10-y)$; rotating $180^\circ$ about $(5,5)$ then gives $(10-x,\,y)$, which is reflection in $x=5$.
b. ii. A single rotation of $90^\circ$ anticlockwise about the centre $(2,5)$. Reason: a translation after a rotation is equivalent to a rotation by the same angle about a shifted centre; here the composition equals rotation $90^\circ$ anticlockwise about $(2,5)$.
❓ EXERCISES
11. The diagram shows five triangles, $A$ to $E$. Here are four transformations.
a. translation $\begin{bmatrix}3\\-1\end{bmatrix}$ followed by reflection in the line $y=3$
b. reflection in the line $x=4$ followed by reflection in the line $y=2$
c. rotation $90^\circ$ anticlockwise, centre $(4,5)$, followed by translation $\begin{bmatrix}-2\\-2\end{bmatrix}$
d. reflection in the line $y=4$ followed by rotation $90^\circ$ anticlockwise, centre $(7,6)$
Work out which triangle is transformed to which other triangle by each of the combined transformations.
👀 Show answer
Method (fill in the table once you read precise coordinates from the grid):
- Record one vertex of each triangle exactly (e.g., $A:(x_A,y_A)$, $B:(x_B,y_B)$, …).
- Apply the card’s transformations to that vertex **in order**:
- Translation$\begin{bmatrix}p\\q\end{bmatrix}$: $(x,y)\mapsto(x+p,\;y+q)$
- Reflection in $y=k$: $(x,y)\mapsto(x,\;2k-y)$; in $x=h$: $(x,y)\mapsto(2h-x,\;y)$
- Rotation$90^\circ$ anticlockwise about $(a,b)$: shift $(X,Y)=(x-a,\;y-b)$, map to $(-Y,\;X)$, then shift back: $(a-Y,\;b+X)$
- Compare the resulting coordinates with the recorded vertices to identify the destination triangle.
- Verify by checking a second vertex from the same triangle.
| Card | Source triangle | Image triangle |
|---|---|---|
| $a$ | ________ | ________ |
| $b$ | ________ | ________ |
| $c$ | ________ | ________ |
| $d$ | ________ | ________ |
Note: With a higher-resolution grid (or listed coordinates of one vertex per triangle), you can complete the table unambiguously using the formulas above.