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{oe}=\overline{at} b) \overline{je}=\overline{ri} c) \angle r=\angle t d) \angle j=\angle a

Explanation:

Step1: Recall properties of translation and reflection

Translation and reflection are rigid motions. Rigid motions preserve side - lengths and angles. So, \(\triangle JOE\cong\triangle BRI\) (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. For example, if \(\triangle JOE\cong\triangle CAT\), \(JO = CA\), \(OE=AT\) is not a result of the congruence from the given rigid - motion sequence.

• Option B:

Since \(\triangle JOE\cong\triangle BRI\) (translation), the corresponding sides are equal. In \(\triangle JOE\) and \(\triangle BRI\), \(\overline{JE}\) and \(\overline{RI}\) are corresponding sides. By the property of translation (a rigid motion), if \(\triangle JOE\) is translated to \(\triangle BRI\), then \(JE = RI\).

• Option C:

In \(\triangle BRI\) and \(\triangle CAT\), \(\angle R\) and \(\angle T\) are not corresponding angles. Since \(\triangle BRI\cong\triangle CAT\) (reflection), corresponding angles are equal. For example, \(\angle B=\angle C\), \(\angle R=\angle A\), \(\angle I=\angle T\).

• Option D:

In \(\triangle JOE\) and \(\triangle CAT\), \(\angle J\) and \(\angle A\) are not corresponding angles. Since \(\triangle JOE\cong\triangle CAT\), corresponding angles are equal. For example, \(\angle J=\angle C\), \(\angle O=\angle T\), \(\angle E=\angle A\).

Answer:

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