Question

1. in the diagram below, △bri is the image of △joe after a translation. triangle cat is the image of △bri after a line reflection. which one of the following statements is always true? a) \\( \overline { o e } = \overline { a t } \\) b) \\( \overline { j e } = \overline { r i } \\) c) \\( \angle r = \angle t \\) d) \\( \angle j = \angle a \\)

Explanation:

Step1: Recall the properties of translation and reflection

Translation and reflection are rigid transformations. Rigid transformations preserve the shape and size of the figure. So, congruent triangles are formed. That is, \(\triangle JOE\cong\triangle BRI\) (because of translation) and \(\triangle BRI\cong\triangle CAT\) (because of reflection). By the transitive property of congruence, \(\triangle JOE\cong\triangle CAT\).

Step2: Analyze each option

• Option A:

In \(\triangle JOE\) and \(\triangle CAT\), \(\overline{OE}\) and \(\overline{AT}\) are not corresponding sides. Since \(\triangle JOE\cong\triangle CAT\), corresponding sides are equal. But \(OE\) corresponds to \(RI\) (from \(\triangle JOE\cong\triangle BRI\)) and \(RI\) corresponds to \(AT\) (from \(\triangle BRI\cong\triangle CAT\)) is not a valid correspondence.

• Option B:

In \(\triangle JOE\) and \(\triangle BRI\) (translation), \(\overline{JE}\) and \(\overline{RI}\) are corresponding sides. Since translation is a rigid transformation, \(\overline{JE}\cong\overline{RI}\).

• Option C:

\(\angle R\) is in \(\triangle BRI\) and \(\angle T\) is in \(\triangle CAT\). They are not corresponding angles. Corresponding angles of congruent triangles are equal. But the correspondence of angles: In \(\triangle BRI\) and \(\triangle CAT\), \(\angle B\) corresponds to \(\angle C\), \(\angle R\) corresponds to \(\angle A\), \(\angle I\) corresponds to \(\angle T\).

• Option D:

\(\angle J\) is in \(\triangle JOE\) and \(\angle A\) is in \(\triangle CAT\). They are not corresponding angles. The correspondence of angles: In \(\triangle JOE\) and \(\triangle CAT\), \(\angle J\) corresponds to \(\angle B\) (from \(\triangle JOE\cong\triangle BRI\)) and \(\angle B\) corresponds to \(\angle C\) (from \(\triangle BRI\cong\triangle CAT\)).

Answer:

B. \(\overline{JE}\cong\overline{RI}\)