Question 9 (12 marks):
(a) Diagram 9.1 shows point P (5, 1) on a Cartesian plane.

Diagram 9.1

Transformation T is a translation $\left(\begin{array}{l}4\\ -3\end{array}\right)$
Transformation S is an enlargement about the centre (–5, 2) with a scale factor 2.
State the coordinates of the image of point P under the following transformations:
(i) T²,
(ii) TS.


(b) Diagram  9.2 shows geometrical shapes KLMNP, KSRQP and KTUVW drawn on a Cartesian plane.

Diagram 9.2

(i) KTUVW is the image of KLMNP under the combined transformation YZ.
Describe, in full, the transformation:
(a) Z,
(b) Y.

(ii) It is given that KSRQP represents a region of area 30 m².
Calculate the area, in m², of the shaded region.


Solution:
(a)


(i) TT = P(–3, 3) → T₁ → P’(1, 0) ) → T₂ → P’’(5, –3)
(ii) TS =  P (–3, 3) → S → P’(–1, 4) → T → P’’(3, 1)


(b)(i)(a)
Z: Reflection in the line x = 0.

(b)(i)(b)
Y: Enlargement with the centre at (0, 0) and a scale factor of 2.

(b)(ii)
Area of KTUVW = (Scale factor)² × Area of object
= 2² × 30
= 120 m²

Hence,
Area of shaded region
= Area KTUVW – Area KSRQP
= 120 – 30
= 90 m²

Question 9 (12 marks):
(a) Diagram 7.1 shows K (5, 1) drawn on a Cartesian plane.
Diagram  7.1

Transformation T is a translation $\left(\begin{array}{l}-3\\ 4\end{array}\right)$
Transformation P is a reflection in the line y = 2.
State the coordinates of the images of point K under each of the following transformations:
(i) T²,
(ii) TP.


(b) Diagram 7.2 shows two pentagons KLMNP and QRSTU, drawn on a Cartesian plane.

Diagram  7.2

(i) Pentagon QRSTU  is the image of pentagon KLMNP under the combined transformation WV.
Describe in full the transformation:
(a) V,
(b) W.

(ii) It is given that pentagon QRSTU represents a region with an area of 90 m².
Calculate the area, in m², of pentagon KLMNP.


Solution:
(a)



(i) TT = K(5, 1) → T₁ → K’(2, 5) ) → T₂ → K’’(–1, 9)
(ii) TP =  K(5, 1) → P → K’(5, 3) → T → K’’(2, 7)


(b)


(b)(i)(a)
V: 90^o rotation in clockwise direction at point K(3, 7).

(b)(i)(b)
W: Enlargement at point (1, 6) with scale factor 3.

(b)(ii)
Area of QRSTU = (scale factor)² × area of object
Area of QRSTU = 3² × Area of KLMNP
90 m² = 9 × Area of KLMNP
Area of KLMNP = 90/9 = 10 m²

Question 8:
(a) Diagram 8.1 shows point A and point B marked on a Cartesian plane.

Diagram 8.1

Transformation R is a rotation of 90^o, clockwise about the centre B.

Transformation T is a translation $\left(\begin{array}{l}5\\ -2\end{array}\right)$

State the coordinates of the image of point A under each of the following transformations:
(i) RT,
(ii) R².

(b) Diagram 8.2 shows three trapeziums ABCD, PQRS and TUVS, drawn on a Cartesian plane.

Diagram 8.2

(i) Trapezium PQRS is the image of trapezium ABCD under the combined transformation MN.
Describe in full, the transformation:
(a) N,  
(b) M.

(ii) It is given that trapezium ABCD represents a region of area 30 m².
Calculate the area, in m², of the shaded region.


Solution:
(a)


(i) A (–1, 6) → T → (4, 4 ) → R → (3, 1)
(ii) A (–1, 6) → R → (5, 6) → R → (5, 0)

(b)(i)(a)
N is a reflection in the line y = 4.

(b)(ii)(b)
M is enlargement of scale factor 2 with centre S (1, 5).

(b)(ii)
Area of PQRS = (scale factor)² x Area of object ABCD
 = 2² x 30
 = 120 m²

Therefore,
Area of shaded region
= Area PQRS – area ABCD
= 120 – 30
= 90 m²

Question 7:
Diagram shows the point J(1, 2) and quadrilaterals ABCD and EFGH, drawn on a Cartesian plane.




(a) Transformation U is a rotation of 90^o, clockwise about the centre O.
Transformation T is a translation $\left(\begin{array}{l}\text{2}\\ 3\end{array}\right)$  
Transformation R is a reflection at the line x = 3.

State the coordinates of the image of point J under each of the following transformations:
(i) RU,
(ii) TR.

(b) EFGH is the image of ABCD under the combined transformation MN.
Describe in full, the transformation:
(i) N,   
(ii) M.

(c) It is given that quadrilateral ABCD represents a region of area 18 m².
Calculate the area, in m², of the shaded region.


Solution:




(a)
(i) J (1, 2) → U → (2, –1 ) → R → (4, –1)
(ii) J (1, 2) → R → (5, 2) → T → (7, 5)

(b)(i)
N is a reflection in the line x = 6.

(b)(ii)
M is enlargement of scale factor 3 with centre (8, 7).

(c)
Area EFGH = (scale factor)² x Area of object ABCD
 = 3² x 18
 = 162 m²

Therefore,
Area of shaded region
= Area of EFGH – Area of ABCD
= 162 – 18
= 144 m²

Question 6:
(a) Diagram below shows point M marked on a Cartesian plane.

Transformation T is a translation $\left(\begin{array}{l}\text{2}\\ 3\end{array}\right)$ and transformation R is an anticlockwise rotation of 90^o about the centre O.
State the coordinates of the image of point M under each of the following transformations:
(i) RT,
(ii) TR,

(b) Diagram below shows two hexagons, ABCDEF and JKLANO, drawn on square grids.



(i) JKLANO is the image of ABCDEF under the combined transformation VW.
Describe in full the transformation:
(a) W   (b) V

(ii) It is given that quadrilateral ABCDEF represents a region of area 45 m².
Calculate the area, in m², of the region represented by the shaded region.


Solution:
(a)

(b)



(i)(a)
V: A reflection in the line EC.

(i)(b)
$\begin{array}{l}\text{Scale factor}=\frac{KL}{BC}=\frac{6}{2}=3\\ \text{W: An enlargement of scale factor 3 at centre }J\text{.}\end{array}$

(ii)
Area of JKLANO
= 3² x area of ABCDEF
= 9 x 45
= 405 m²

Area of the coloured region
= area of JKLANO – area of ABCDEF
= 405 – 45
= 360 m²

Question 5:
(a) Diagram below shows two points, M and N, on a Cartesian plane.



Transformation T is a translation $\left(\begin{array}{l}3\\ -1\end{array}\right)$ and transformation R is an anticlockwise rotation of 90^o about the centre (0, 2).
(i) State the coordinates of the image of point M under transformation R.
(ii) State the coordinates of the image of point N under the following transformations:
(a) T²,
(b) TR,

(b) Diagram below shows three pentagons, A, B and C, drawn on a Cartesian plane.


(i) C is the image of A under the combined transformation WV.
Describe in full the transformation:
(a) V    (b) W
(ii) Given A represents a region of area 12 m², calculate the area, in m², of the region represented by C.


Solution:
(a)



(b)


(b)(i)(a)
V: A reflection in the line x  = 8

(b)(i)(b)
W: An enlargement of scale factor 2 with centre (14, 0).

(b)(ii)
Area of B = area of A = 12 m²
Area of C = (Scale factor)² x Area of object
 = 2² x area of B
 = 2² x 12
 = 48 m²

Question 4:
Diagram below shows three triangles RPQ, UST and RVQ, drawn on a Cartesian plane.


(a) Transformation R is a rotation of 90^o, clockwise about the centre O.
Transformation T is a translation $\left(\begin{array}{l}2\\ \text{3}\end{array}\right)$ .
State the coordinates of the image of point B under each of the following transformations:
(i) Translation T²,
(ii) Combined transformation TR.

(b)
(i) Triangle UST is the image of triangle RPQ under the combined transformation VW.
Describe in full the transformation:
(a) W   (b) V

(ii) It is given that quadrilateral RPQ represents a region of area 15 m².
Calculate the area, in m², of the region represented by the shaded region.


Solution:


(a)
(i) (–5, 3) → T → (–3, 6) ) → T → (–1, 9)
(ii) (–5, 3) → R → (3, 5) → T → (5, 8)

(b)(i)(a)
W: A reflection in the line URQT.

(b)(i)(b)
$\begin{array}{l}\text{Scale factor}=\frac{US}{RV}=\frac{6}{2}=3\\ \text{V: An enlargement of scale factor 3 at centre }\left(-4,2\right)\end{array}$

(b)(ii)
Area of UST = (Scale factor)² x Area of RPQ
= 3² x area of RPQ
= 3² x 15
= 135 m²

Therefore,
Area of the shaded region
= Area of UST – area of RPQ
= 135 – 15
= 120 m²