GCSE-level math worksheet focusing on combining geometric transformations, including reflections, rotations, translations, and enlargements on coordinate grids.

Here are the step-by-step solutions to find the coordinates for each transformed shape.

1) Shape A to Shape B

Transformations: Reflect in the y-axis, then translate by vector $\begin{pmatrix} -2 \\ -6 \end{pmatrix}$.

* Original Coordinates (Shape A): $(-5, 5), (-3, 5), (-3, 1), (-4, 1), (-4, 3), (-5, 3)$.
* Step 1: Reflect in y-axis. Rule: $(x, y) \rightarrow (-x, y)$.
* $(-5, 5) \rightarrow (5, 5)$
* $(-3, 5) \rightarrow (3, 5)$
* $(-3, 1) \rightarrow (3, 1)$
* $(-4, 1) \rightarrow (4, 1)$
* $(-4, 3) \rightarrow (4, 3)$
* $(-5, 3) \rightarrow (5, 3)$
* Step 2: Translate by $\begin{pmatrix} -2 \\ -6 \end{pmatrix}$. Rule: Subtract 2 from x, subtract 6 from y.
* $(5, 5) \rightarrow (3, -1)$
* $(3, 5) \rightarrow (1, -1)$
* $(3, 1) \rightarrow (1, -5)$
* $(4, 1) \rightarrow (2, -5)$
* $(4, 3) \rightarrow (2, -3)$
* $(5, 3) \rightarrow (3, -3)$

Shape B Vertices: $(3, -1), (1, -1), (1, -5), (2, -5), (2, -3), (3, -3)$.

---

2) Triangle C to Triangle D

Transformations: Rotate $180^\circ$ about $(-1, 2)$, then reflect in line $y = x$.

* Original Coordinates (Triangle C): $(0, 3), (0, 0), (4, 0)$.
* Step 1: Rotate $180^\circ$ about $(-1, 2)$. Rule: The center is the midpoint. New point $P' = 2(\text{Center}) - P$.
* $(0, 3) \rightarrow 2(-1, 2) - (0, 3) = (-2, 4) - (0, 3) = (-2, 1)$
* $(0, 0) \rightarrow 2(-1, 2) - (0, 0) = (-2, 4) - (0, 0) = (-2, 4)$
* $(4, 0) \rightarrow 2(-1, 2) - (4, 0) = (-2, 4) - (4, 0) = (-6, 4)$
* Step 2: Reflect in line $y = x$. Rule: Swap x and y coordinates $(x, y) \rightarrow (y, x)$.
* $(-2, 1) \rightarrow (1, -2)$
* $(-2, 4) \rightarrow (4, -2)$
* $(-6, 4) \rightarrow (4, -6)$

Triangle D Vertices: $(1, -2), (4, -2), (4, -6)$.

---

3) Shape E to Shape F

Transformations: Enlarge by scale factor -2 from center $(2, -3)$, then translate by vector $\begin{pmatrix} -4 \\ 3 \end{pmatrix}$.

* Original Coordinates (Shape E): $(1, -6), (4, -6), (4, -5), (3, -5), (3, -3), (2, -3)$.
* Step 1: Enlarge by SF -2 from $(2, -3)$. Rule: Vector from center is multiplied by -2.
* $(1, -6)$: Vector is $(-1, -3)$. New vector $(2, 6)$. Point: $(2+2, -3+6) = (4, 3)$.
* $(4, -6)$: Vector is $(2, -3)$. New vector $(-4, 6)$. Point: $(2-4, -3+6) = (-2, 3)$.
* $(4, -5)$: Vector is $(2, -2)$. New vector $(-4, 4)$. Point: $(2-4, -3+4) = (-2, 1)$.
* $(3, -5)$: Vector is $(1, -2)$. New vector $(-2, 4)$. Point: $(2-2, -3+4) = (0, 1)$.
* $(3, -3)$: Vector is $(1, 0)$. New vector $(-2, 0)$. Point: $(2-2, -3+0) = (0, -3)$.
* $(2, -3)$: This is the center, so it stays $(2, -3)$.
* Step 2: Translate by $\begin{pmatrix} -4 \\ 3 \end{pmatrix}$. Rule: Subtract 4 from x, add 3 to y.
* $(4, 3) \rightarrow (0, 6)$
* $(-2, 3) \rightarrow (-6, 6)$
* $(-2, 1) \rightarrow (-6, 4)$
* $(0, 1) \rightarrow (-4, 4)$
* $(0, -3) \rightarrow (-4, 0)$
* $(2, -3) \rightarrow (-2, 0)$

Shape F Vertices: $(0, 6), (-6, 6), (-6, 4), (-4, 4), (-4, 0), (-2, 0)$.

---

4) Shape G to Shape H

Transformations: Rotate $90^\circ$ clockwise about origin, then reflect in line $x = -3$.

* Original Coordinates (Shape G): $(-5, -3), (-2, -3), (-2, -5), (-4, -5)$.
* Step 1: Rotate $90^\circ$ clockwise about $(0,0)$. Rule: $(x, y) \rightarrow (y, -x)$.
* $(-5, -3) \rightarrow (-3, 5)$
* $(-2, -3) \rightarrow (-3, 2)$
* $(-2, -5) \rightarrow (-5, 2)$
* $(-4, -5) \rightarrow (-5, 4)$
* Step 2: Reflect in line $x = -3$. Rule: Distance from line is preserved. Formula: $x_{new} = 2(-3) - x_{old} = -6 - x_{old}$.
* $(-3, 5) \rightarrow (-6 - (-3), 5) = (-3, 5)$
* $(-3, 2) \rightarrow (-6 - (-3), 2) = (-3, 2)$
* $(-5, 2) \rightarrow (-6 - (-5), 2) = (-1, 2)$
* $(-5, 4) \rightarrow (-6 - (-5), 4) = (-1, 4)$

Shape H Vertices: $(-3, 5), (-3, 2), (-1, 2), (-1, 4)$.

---

5) Triangle I to Triangle J

Transformations: Enlarge by scale factor $\frac{1}{2}$ from center $(0, -1)$, then reflect in line $y = -x$.

* Original Coordinates (Triangle I): $(-2, 5), (4, 5), (4, 1)$.
* Step 1: Enlarge by SF $\frac{1}{2}$ from $(0, -1)$. Rule: Vector from center is halved.
* $(-2, 5)$: Vector is $(-2, 6)$. Half is $(-1, 3)$. Point: $(0-1, -1+3) = (-1, 2)$.
* $(4, 5)$: Vector is $(4, 6)$. Half is $(2, 3)$. Point: $(0+2, -1+3) = (2, 2)$.
* $(4, 1)$: Vector is $(4, 2)$. Half is $(2, 1)$. Point: $(0+2, -1+1) = (2, 0)$.
* Step 2: Reflect in line $y = -x$. Rule: Swap and negate $(x, y) \rightarrow (-y, -x)$.
* $(-1, 2) \rightarrow (-2, 1)$
* $(2, 2) \rightarrow (-2, -2)$
* $(2, 0) \rightarrow (0, -2)$

Triangle J Vertices: $(-2, 1), (-2, -2), (0, -2)$.

---

6) Shape K to Shape L

Transformations: Translate by vector $\begin{pmatrix} 0 \\ -4 \end{pmatrix}$, then rotate $90^\circ$ anti-clockwise about $(-1, 0)$.

* Original Coordinates (Shape K): $(-3, 3), (-1, 7), (0, 3), (-2, 1)$.
* Step 1: Translate by $\begin{pmatrix} 0 \\ -4 \end{pmatrix}$. Rule: Keep x, subtract 4 from y.
* $(-3, 3) \rightarrow (-3, -1)$
* $(-1, 7) \rightarrow (-1, 3)$
* $(0, 3) \rightarrow (0, -1)$
* $(-2, 1) \rightarrow (-2, -3)$
* Step 2: Rotate $90^\circ$ anti-clockwise about $(-1, 0)$. Rule: Relative vector $(dx, dy)$ becomes $(-dy, dx)$.
* $(-3, -1)$: Rel to $(-1,0)$ is $(-2, -1)$. Rotated is $(1, -2)$. Abs: $(-1+1, 0-2) = (0, -2)$.
* $(-1, 3)$: Rel to $(-1,0)$ is $(0, 3)$. Rotated is $(-3, 0)$. Abs: $(-1-3, 0+0) = (-4, 0)$.
* $(0, -1)$: Rel to $(-1,0)$ is $(1, -1)$. Rotated is $(1, 1)$. Abs: $(-1+1, 0+1) = (0, 1)$.
* $(-2, -3)$: Rel to $(-1,0)$ is $(-1, -3)$. Rotated is $(3, -1)$. Abs: $(-1+3, 0-1) = (2, -1)$.

Shape L Vertices: $(0, -2), (-4, 0), (0, 1), (2, -1)$.

Final Answer:
1) Shape B: Draw a polygon connecting $(3, -1), (1, -1), (1, -5), (2, -5), (2, -3), (3, -3)$.
2) Triangle D: Draw a triangle connecting $(1, -2), (4, -2), (4, -6)$.
3) Shape F: Draw a polygon connecting $(0, 6), (-6, 6), (-6, 4), (-4, 4), (-4, 0), (-2, 0)$.
4) Shape H: Draw a trapezium connecting $(-3, 5), (-3, 2), (-1, 2), (-1, 4)$.
5) Triangle J: Draw a triangle connecting $(-2, 1), (-2, -2), (0, -2)$.
6) Shape L: Draw a kite/quadrilateral connecting $(0, -2), (-4, 0), (0, 1), (2, -1)$.