A rectangle `A B C D ,` where `A-=(0,0),B-=(4,0),C-=(4,2)D-=(0,2)` , undergoes the following transformations successively: `f_1(x ,y)vec(y ,x)` `f_2(x , y)vec(x+3y ,y)` `f_3(x , y)vec((x-y)//2),(x+y)//2)` The final figure will be square (b) a rhombus a rectangle (d) a parallelogram

Options

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

Answer

D

Explanation

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)=(8)/(4)m_(BC)=(9-8)/(5-4)=1, m_(CD)=(9-1)/(5-1)=(8)/(4)`,
`m_(AD)=1,m_(BC)=(9)/(5), m_(CD)=(8-1)/(4-1)=(7)/(3)`,
Hence, the final figure will be a parallelogram.