Question

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

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

Answer 1

Answer: (D)

Clearly, $A$ will remain as $(0,0)$ ;
$f_1$ will make $B$ as $(0,4)$,
$f_2$ will make it $(12,4)$,
and $f_3$ will make it $(4,8)$;
$f_1$ will make $C$ as $(2,4)$,
$f_2$ will make it $(14,4)$,
and $f_3$ will make it $(5,9)$.
Finally, $f_1$ will make $D$ as $(2,0)$,
$f_2$ will make it $(2,0)$, and
$f_3$ will make it $(1,1)$.
So, we finally get $A(0,0),B(4,8),C(5,9)$, and $D(1,1)$.
Hence, $m_{AB}=\frac{8}{4}$,
$m_{BC}=\frac{9-8}{5-4}=1$,
$m_{CD}=\frac{9-1}{5-1}=\frac{8}{4}$,
$m_{AD}=1$,
$m_{AC}=\frac{9}{5}$,
$m_{BD}=\frac{8-1}{4-1}=\frac{7}{3}$
Hence, the final figure will be a parallelogram.