Question

consider the lines shown below.
which statements are correct?
select all that apply.
a. if both lines are translated 6 units down, they will be parallel to each other.
b. if both lines are reflected across the y - axis, they will be parallel to each other.
c. if both lines are translated 3 units to the right, they will be perpendicular to each other.
d. if both lines are rotated 180° counterclockwise about the origin, they will be parallel to each other.
e. if both lines are rotated 90° counterclockwise about the origin, they will be perpendicular to each other.

Explanation:

Step1: Recall the property of parallel lines

Parallel lines have the same slope. Transformations like translation (shifting), reflection, and rotation (by \(180^{\circ}\)) do not change the slope of a line.

• For option A: Translating both lines 6 units down is a vertical translation. The slope of each line remains the same. So, they will still be parallel.
• For option B: Reflecting both lines across the \(y -\)axis. The formula for reflecting a point \((x,y)\) across the \(y -\)axis is \((-x,y)\). For a line \(y = mx + b\), after reflection \(y=-mx + b\) (if \(m\) is the slope of the original line). But if two original lines have the same slope \(m_1=m_2\), after reflection \(y=-m_1x + b_1\) and \(y=-m_2x + b_2\), and \(-m_1=-m_2\). So, they are parallel.
• For option C: Translating both lines 3 units to the right is a horizontal translation. The slope of each line remains the same. Parallel lines remain parallel (slope does not change for translation), not perpendicular.
• For option D: Rotating a line by \(180^{\circ}\) counter - clockwise about the origin. The formula for rotating a point \((x,y)\) by \(180^{\circ}\) counter - clockwise about the origin is \((-x,-y)\). For a line \(y = mx + b\), if \((x_1,y_1)\) lies on the line \(y_1=mx_1 + b\), after rotation \((-x_1,-y_1)\) lies on the new line \(-y_1=m(-x_1)+b'\), or \(y_1 = mx_1 - b'\). The slope \(m\) remains the same. So, two parallel lines (same slope initially) after \(180^{\circ}\) rotation have the same slope.
• For option E: Rotating a line by \(90^{\circ}\) counter - clockwise about the origin. The slope \(m\) of a line changes to \(-\frac{1}{m}\). If two original lines are parallel (same \(m\)), after rotation, their slopes are \(-\frac{1}{m}\) (same value). They are parallel, not perpendicular.

Answer:

A. If both lines are translated 6 units down, they will be parallel to each other; B. If both lines are reflected across the y - axis, they will be parallel to each other; D. If both lines are rotated \(180^{\circ}\) counterclockwise about the origin, they will be parallel to each other.