🔄✨ Mastering Combined Transformations!

Transformations such as translations, reflections, rotations and enlargements are the "moves" that allow us to reshape and reposition figures in coordinate geometry. **Combining two or more of these moves in the correct order** is a frequent examination task because it checks both your understanding of each individual transformation and your ability to track points methodically. Nail this skill and you will unlock quick marks in paper-1 as well as in structured problems on paper-2!

Main Concept: Order Matters 🚦

When asked to perform, say, a rotation followed by a translation, remember that the first transformation acts on the *original* object, while the second acts on the *image* produced by the first. A quick way to keep control is to write coordinate mappings. For instance, a translation by

\begin{pmatrix}a\\b\end{pmatrix}

takes any point

(x,y)

(x+a,\,y+b)

, whereas a

90^\circ

anticlockwise rotation about the origin maps

(x,y)

(-y,\,x)

. **Write these mappings one underneath the other** so you can substitute neatly and avoid sign errors.

Worked Example: Rotate then Translate

Problem: Triangle

PQR

has vertices

P(2,1)

Q(5,1)

and

R(2,4)

. It is first rotated

90^\circ

anticlockwise about the origin and then translated by

\begin{pmatrix}3\\-2\end{pmatrix}

. Find the coordinates of

P',Q',R'

after both transformations. Solution:
Step 1: Apply the rotation to each vertex. The mapping

(x,y) \to (-y,\,x)

gives:

P(2,1) \to P_1(-1,2),\; Q(5,1) \to Q_1(-1,5),\; R(2,4) \to R_1(-4,2).

Step 2: Now translate each rotated point by

\begin{pmatrix}3\\-2\end{pmatrix}

. This mapping

(x,y) \to (x+3,\,y-2)

yields:

P'(-1+3,\,2-2)=(2,0),\; Q'(-1+3,\,5-2)=(2,3),\; R'(-4+3,\,2-2)=(-1,0).

Therefore the final image has vertices

P'(2,0)

Q'(2,3)

and

R'(-1,0)

. **Practise chaining mappings like this and you will move confidently through any combined-transformation question!**