Question

A rectangle ABCD, 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) $f_3(x,y)\to (\frac{x-y}{2},\frac{x+y}{2})$
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

A remains (0, 0)
$f_1\left(B\right)=(0,4),f_2\left(f_1\right(B\left)\right)=(12,4)$, and $f_3\left(f_2\right(f_1\left(B\right)\left)\right)=(\frac{12-4}{2},\frac{12+4}{2})=(4,8)$
So finally B = (4, 8)
$C=(4,2)\stackrel{f_1}{\to }(2,4)\stackrel{f_2}{\to }(14,4)\stackrel{f_3}{\to }(5,9)D=(0,2)\stackrel{f_1}{\to }(2,0)\stackrel{f_2}{\to }(2,0)\stackrel{f_3}{\to }(1,1)$
So Slope of AB $=\frac{8}{4}=2$
Slope of CD $=\frac{9-1}{5-1}=2$
Slope of BC = 1
Slope of AD = 1
So, AB is parallel to DC, BC is parallel to AD. Length of AB $\ne$ length of BC, AB is not perpendicular to BC.
Hence ABCD is a parallelogram.