Question

1. which one of the following transformations of \\( \overline { o a } \\) would result in an image parallel to \\( \overline { o a } \\)?

a) a clockwise rotation of \\( 90 ^ { \circ } \\) about the origin
b) a reflection over the \\( y \\)-axis
c) a translation of two units down
d) a reflection over the \\( x \\)-axis

Explanation:

Step1: Recall transformation properties

• Rotation changes the direction (unless \(360^{\circ}\) rotation). A \(90^{\circ}\) clock - wise rotation (Option A) will change the slope of \(\overline{OA}\), so the image will not be parallel.
• Reflection (over \(x\) - axis (Option D) or \(y\) - axis (Option B)) changes the orientation. For a reflection over \(y\) - axis, if \(A=(x,y)\), its image is \((-x,y)\). For a reflection over \(x\) - axis, if \(A=(x,y)\), its image is \((x, - y)\). In both cases, the slope of the line segment changes.
• Translation (Option C): A translation (shifting) of a line segment. The slope of a line segment is preserved under translation. If we have a line segment \(\overline{OA}\) with slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\) (where \(O=(0,0)\) and \(A=(x,y)\)), after a translation \(T:(x,y)\to(x,y - 2)\) (translation of two units down), the new line segment (with endpoints \(O'=(0,- 2)\) and \(A'=(x,y - 2)\)) has slope \(m'=\frac{(y - 2)-(-2)}{x-0}=\frac{y}{x}\), which is the same as the slope of \(\overline{OA}\).

Answer:

C. a translation of two units down