Question

reflecting \\( \triangle l m n \\) across the horizontal line \\( y = 1 \\), we get its image \\( \triangle l ^ { prime } m ^ { prime } n ^ { prime } \\). suppose \\( \overline { l l ^ { prime } } \\), \\( \overline { m m ^ { prime } } \\), and \\( \overline { n n ^ { prime } } \\) intersect the line of reflection at \\( s, t \\), and \\( u \\) as shown below. complete the following. (a) select all that apply. \\( \overline { l l ^ { prime } } \\) is perpendicular to \\( \overline { m m ^ { prime } } \\) and \\( \overline { n n ^ { prime } } \\). \\( \overline { l l ^ { prime } } \\), \\( \overline { m m ^ { prime } } \\), and \\( \overline { n n ^ { prime } } \\) are each perpendicular to the line of reflection. \\( \overline { l l ^ { prime } } \\), \\( \overline { m m ^ { prime } } \\), and \\( \overline { n n ^ { prime } } \\) are parallel to each other. none of the above (b) suppose each pair of segments below have the same length. find each length. \\( l s = l ^ { prime } s = \square \\) units \\( m t = m ^ { prime } t = \square \\) units \\( n u = n ^ { prime } u = \square \\) units (c) choose the correct statement about the reflection. the line of reflection is the perpendicular bisector of each segment joining a point and its image. the line of reflection is parallel to each segment joining a point and its image. the line of reflection is neither parallel nor perpendicular to each segment joining a point and its image. each side of the original figure is perpendicular to its image.

Explanation:

Step1: Analyze part (a)

• For a reflection across a line \(y = 1\), the segments joining a point and its image (\(\overline{LL'}\), \(\overline{MM'}\), \(\overline{NN'}\)) are perpendicular to the line of reflection. Also, since they are all perpendicular to the same line (\(y = 1\)), they are parallel to each other.
• \(\overline{LL'}\) is not perpendicular to \(\overline{MM'}\) and \(\overline{NN'}\) because they are all parallel (as they are all perpendicular to the line \(y = 1\)).

Step2: Analyze part (b)

• Count the units from \(L\) to \(S\) (or \(L'\) to \(S\)). From the graph, \(LS = L'S=5\) units.
• Count the units from \(M\) to \(T\) (or \(M'\) to \(T\)). From the graph, \(MT = M'T = 6\) units.
• Count the units from \(N\) to \(U\) (or \(N'\) to \(U\)). From the graph, \(NU = N'U = 4\) units.

Step3: Analyze part (c)

• By the property of reflection, the line of reflection is the perpendicular bisector of each segment joining a point and its image.

Answer:

(a) \(\overline{LL'},\overline{MM'},\) and \(\overline{NN'}\) are each perpendicular to the line of reflection; \(\overline{LL'},\overline{MM'},\) and \(\overline{NN'}\) are parallel to each other.
(b) \(LS = L'S = 5\) units; \(MT = M'T = 6\) units; \(NU = N'U = 4\) units.
(c) The line of reflection is the perpendicular bisector of each segment joining a point and its image.