Question

right triangle abc is reflected over \\( \overline { a c } \\), then dilated by a scale factor of \\( \frac { 2 } { 3 } \\) to form triangle dec. which statements about the two triangles must be true? select three options. \\( \square \triangle a b c \sim \triangle d e c \\) \\( \square \angle b \cong \angle e \\) \\( \square 3 b c = 2 e c \\) \\( \square 3 d e = 2 a b \\) \\( \square 3 m \angle a = 2 m \angle d \\) \\( \square 2 m \angle a = 3 m \angle d \\)

Explanation:

Step1: Properties of reflection and dilation

Reflection preserves shape and size (congruence), dilation preserves shape (similarity). So \(\triangle ABC\sim\triangle DEC\) (by AA similarity: right - angles and common angle at \(C\)).

Step2: Corresponding angles in similar triangles

Since \(\triangle ABC\sim\triangle DEC\), corresponding angles are congruent. But also, from the dilation factor \(\frac{2}{3}\), if \(\triangle ABC\) is the original and \(\triangle DEC\) is the dilated triangle (\(\triangle DEC\) is the image of \(\triangle ABC\) after reflection and dilation), for side lengths: \(\frac{DE}{AB}=\frac{EC}{BC}=\frac{2}{3}\). Cross - multiplying \(\frac{DE}{AB}=\frac{2}{3}\) gives \(3DE = 2AB\), and cross - multiplying \(\frac{EC}{BC}=\frac{2}{3}\) gives \(2BC=3EC\) (not \(3BC = 2EC\)). Angles in similar triangles are equal, so \(m\angle A=m\angle D\). Also, since \(\triangle ABC\sim\triangle DEC\), \(\angle B\) and \(\angle E\) are corresponding angles.

Answer:

\(\triangle ABC\sim\triangle DEC\), \(\angle B\cong\angle E\), \(3DE = 2AB\)