Question

A rectangle ABCD, A $\equiv$ (0,0), B $\equiv$ (4,0),C $\equiv$ (4,2) and D $\equiv$ (0,2) undergoes the following transformations successively:
(i) $f_1$(x, y) $\to$ (y, x)
(ii) $f_2$(x, y) $\to$ (x +3y, y)
(iii) $f_3$(x, y) $\to$$(\frac{x-y}{2},\frac{x+y}{2})$
The final figure will be

• A. a square
• B. a rhombus
• C. a rectangle
• D. a parallelogram

Answer 1

The correct option is D a parallelogram

Given Points are $A\equiv (0,0),B(4,0),C(4,2),D(0,2)$

After first transformation neq points are

$A\equiv (0,0),B\equiv (0,4),C\equiv (2,4),D\equiv (2,0)$

After $2^{nd}$ transformation

i.e. $x\to x+3y$ & $y\to y$

$\therefore A\equiv (0,0),B\equiv (12,4),C\equiv (14,4),D\equiv (2,0)$

$\uparrow$ $\uparrow$ $\uparrow$ $\uparrow$

$0+3\left(0\right)$ $0+3\left(4\right)$ $2+3\left(4\right)$ $2+3\left(0\right)$

After $3^{rd}$ transformation

i.e $x\to \frac{x-y}{2},y\to \frac{x+y}{2}$

Points becomes

$A\equiv (0,0),B\equiv (\frac{12-4}{2},\frac{12+4}{2}),C\equiv (\frac{14-4}{2},\frac{14+4}{2}),D\equiv (\frac{2-0}{2},\frac{2+0}{2})$

$B\equiv (4,8)$ $C\equiv (5,9)$ $D\equiv (1,1)$

$\therefore$ From rectangle $ABCD$ we get a parallelogram