Question

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:

• Translation: Translation (shifting) of a line does not change its slope. If two lines are parallel initially, translating them (either down or to the right) will keep them parallel. So, if two lines are parallel, translating them 6 units down (Option A) will keep them parallel. Translating them 3 units to the right (Option C) will not make them perpendicular (since their slopes remain the same as the original parallel lines).
• Reflection: Reflection across the \(y -\)axis changes the \(x\) - coordinate of points on the line. But for two parallel lines, reflection across the \(y -\)axis will result in two new lines that are still parallel. The slope of a line \(y = mx + b\) reflected across the \(y -\)axis is \(y=-mx + b\). If two original lines \(y = m_1x + b_1\) and \(y=m_2x + b_2\) with \(m_1=m_2\), after reflection \(y=-m_1x + b_1\) and \(y =-m_2x + b_2\) are still parallel (\(-m_1=-m_2\)).
• Rotation:
• A \(180^{\circ}\) counter - clockwise rotation about the origin of a line \(y = mx + b\) (using the rotation formula \((x,y)\to(-x,-y)\)) gives a line with the same slope. If two lines are parallel (\(m_1 = m_2\)) initially, after \(180^{\circ}\) rotation, their slopes remain equal.
• A \(90^{\circ}\) counter - clockwise rotation about the origin of a line \(y=mx + b\) (using the rotation formula \((x,y)\to(-y,x)\)) changes the slope from \(m\) to \(-\frac{1}{m}\). If two lines are parallel (\(m_1=m_2\)) initially, after \(90^{\circ}\) rotation, their slopes \(-\frac{1}{m_1}\) and \(-\frac{1}{m_2}\) are equal (not perpendicular, since for perpendicular lines \(m_1\times m_2=- 1\))

Answer:

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.