SPM Practice (Long Questions)

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