Question

transformations test (topic 4) consider the line segment defined by points a(0,1) and b(4,6). how does a reflection across the x-axis affect \\(\overline{ab}\\)? select two. the y-values of the endpoints become their opposites. the length of the reflection of \\(\overline{ab}\\) is less than the length of \\(\overline{ab}\\). the length of the reflection of \\(\overline{ab}\\) is greater than the length of \\(\overline{ab}\\). the length of the reflection of \\(\overline{ab}\\) is the same as the length of \\(\overline{ab}\\). the x-values of the reflection are the opposite values of the x-values of the original segment.

Explanation:

1. For reflection across the x - axis, the transformation rule is \((x,y)\to(x, - y)\). So for points \(A(0,1)\) and \(B(4,6)\), after reflection, \(A\) becomes \(A'(0, - 1)\) and \(B\) becomes \(B'(4,-6)\). This shows that the \(y\) - values of the endpoints become their opposites.
2. Reflection is a rigid transformation. Rigid transformations (like reflection, translation, rotation) preserve the length of the figure. So the length of the reflection of \(\overline{AB}\) is the same as the length of \(\overline{AB}\).

• The statement "The length of the reflection of \(\overline{AB}\) is less than the length of \(\overline{AB}\)" is wrong because reflection preserves length.
• The statement "The length of the reflection of \(\overline{AB}\) is greater than the length of \(\overline{AB}\)" is wrong for the same reason (reflection is rigid).
• The statement "The \(x\) - values of the reflection are the opposite values of the \(x\) - values of the original segment" is wrong because in reflection across the \(x\) - axis, \(x\) - values remain the same.

Answer:

• The \(y\) - values of the endpoints become their opposites.
• The length of the reflection of \(\overline{AB}\) is the same as the length of \(\overline{AB}\).