SPM Practice (Long Questions)

Question 2:
(a) Diagram below shows point A and straight line y + x = 5 drawn on a Cartesian plane.


Transformation T is a translation $\left(\begin{array}{l}5\\ -2\end{array}\right)$
Transformation R is a reflection at the line y + x = 5.

State the coordinates of the image of point A under each of the following transformations:
(i) Transformation T,
(ii) Combined transformation TR.

(b) Diagram below shows pentagons JKLMN, PQRST and PUVWX, drawn on a Cartesian plane.

(i) PUVWX is the image of JKLMN under the combined transformation CB.
Describe in full the transformation:
(a) B    (b) C

(ii) It is given that pentagon JKLMN represents a region of area 80 m² .
Calculate the area, in m² , of the region represented by the shaded region.

Solution:
(a)

(i) (3, 4) → T → (8, 2)
(ii) (3, 4) → R → (1, 2) → T → (6, 0)

(b)

(b)(i)(a)
B: A clockwise rotation of  90^o about the centre (0, 2).

(b)(i)(b)
$\begin{array}{l}\text{Scale factor}=\frac{PU}{PQ}=\frac{6}{4}=\frac{3}{2}\\ \text{C: An enlargement of scale factor }\frac{3}{2}\text{ with centre }P\left(2,0\right)\end{array}$

(b)(ii)
Area of PQRST = Area of JKLMN
= 80 m²

$\begin{array}{l}\text{Area of }PUVWX\\ ={\left(\frac{3}{2}\right)}^2\times \text{area of }PQRST\\ =\frac{9}{4}\times 80\\ =180{\text{ m}}^2\\ \therefore \text{ Area of the shaded region}\\ \text{= area of }PUVWX-\text{area of }PQRST\\ =180-80\\ =100{\text{ m}}^2\end{array}$