First slide

Equation of line in Straight lines

Question

$A rectangle A B C D, where A \equiv (0, 0), B \equiv (4, 0), C \equiv (4, 2), D \equiv (0, 2), undergoes the following transformations successively:$
$i. f_{1} (x, y) \to (y, x)$
$ii. f_{2} (x, y) \to (x + 3 y, y)$
$iii. f_{3} (x, y) \to ((x - y) / 2, (x + y) / 2)$
The final figure will be

Moderate

A

a square
B

a rhombus
C

a rectangle
D

a parallelogram

Solution

$Clearly, A will remain as (0, 0); f_{1} will make B as (0, 4), f_{2} will make it (12, 4), and f_{3} will make it (4, 8); f_{1} will$
$make C as (2, 4), f_{2} will make it (14, 4), and f_{3} will make it (5, 9) . Finally, f_{1} will make D as (2, 0), f_{2} will make it (2, 0), and$
$f_{3} will make it (1, 1) . So, we finally get A \equiv (0, 0), B \equiv (4, 8), C \equiv (5, 9), and D \equiv (1, 1) . Hence,$
$\begin{aligned} m_{A B} = \frac{8}{4}, m_{B C} = \frac{9 - 8}{5 - 4} = 1, m_{C D} = \frac{9 - 1}{5 - 1} = \frac{8}{4}, \\ m_{A D} = 1, m_{A C} = \frac{9}{5}, m_{B D} = \frac{8 - 1}{4 - 1} = \frac{7}{3} \end{aligned}$
Hence, the final figure will be a parallelogram.

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