Question

A rectangle $A B C D$, where $A(0,0), B(4,0), C(4,2)$, $D(0,2)$, undergoes the following transformations successively
i. $f_{1}(x, y) \rightarrow(y, x)$
ii. $f_{2}(x, y) \rightarrow(x+3 y, y)$
iii. $f_{3}(x, y) \rightarrow((x-y) / 2,(x+y) / 2)$
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, $m_{A B}=\frac{8}{4}$,
$ m_{B C}=\frac{9-8}{5-4}=1$,
$ m_{C D}=\frac{9-1}{5-1}=\frac{8}{4}$,
$m_{A D}=1$,
$ m_{A C}=\frac{9}{5}$,
$ m_{B D}=\frac{8-1}{4-1}=\frac{7}{3}$
Hence, the final figure will be a parallelogram.