Answer: The correct option is (C) Rotated parallel lines always remain parallel lines.
Step-by-step explanation: We are given ton select the correct statement about the rotation of two parallel lines.
We know that two parallel lines have same slope.
So, let m = 2 be the slope of the two parallel lines and 4, 7 are the y-intercepts.
Then, the equations of the lines are given by
\(y=2x+4~~~~~~~~~~~~~~~~~~~~(i)\\y=2x+7~~~~~~~~~~~~~~~~~~~~(ii)\)
Let us rotate the lines through an angle of 90° clockwise.
The co-ordinates of the point (x, y), after rotating 90° clockwise becomes (y, -x).
We have
P(-1, 2) and Q(1, 6) are two points on the line (i) and R(-1, 5) and S(1, 9) are two points on the line (ii).
So, after rotation, they become
P(-1, 2) ⇒ P'(2, 1),
Q(1, 6) ⇒ Q'(6, -1),
R(-1, 5) ⇒ R'(5, 1),
S(1, 9) ⇒ S'(9, -1).
So, slope of line (i) is
\(m_1=\dfrac{-1-1}{6-2}=-\dfrac{1}{2},\)
and the slope of line (ii) is
\(m_2=\dfrac{-1-1}{9-5}=-\dfrac{1}{2}.\)
Since, \(m_1=m_2,\)
so the slopes of the lines after rotation is again same, and hence they are again parallel to each other.
So, the angle of rotation (90° or 180° or 360°)does not matter, the parallel lines will always be parallel.
Thus, the rotated parallel lines always remain parallel lines.
Option (C) is correct.
