9.6 SPM Practice (Long Questions)

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