Question

1. tanya performs two transformations on \\( \triangle a b c \\) to form \\( \triangle a^{\prime \prime} b^{\prime \prime} c^{\prime \prime} \\), as shown on the coordinate grid below. determine whether each statement about the triangles is true. choose true or false for each statement. \\( \triangle a b c \\) is similar to \\( \triangle a^{\prime \prime} b^{\prime \prime} c^{\prime \prime} \\). true false \\( \triangle a b c \\) is congruent to \\( \triangle a^{\prime \prime} b^{\prime \prime} c^{\prime \prime} \\). true fal \\( \triangle a b c \\) can be reflected across the y - axis to form \\( \triangle a^{\prime} b^{\prime} c^{\prime} \\), and then \\( \triangle a^{\prime} b^{\prime} c^{\prime} \\) can be rotated \\( 90^{\circ} \\) clockwise about the origin to form \\( \triangle a^{\prime \prime} b^{\prime \prime} c^{\prime \prime} \\). true false \\( \triangle a b c \\) can be rotated \\( 90^{\circ} \\) counterclockwise about the origin to form \\( \triangle a^{\prime} b^{\prime} c^{\prime} \\), and then \\( \triangle a^{\prime} b^{\prime} c^{\prime} \\) can be dilated by a scale factor of 2 with the center of dilation at the origin

Explanation:

Step1: Similarity and Congruence Concepts

Similar triangles have the same shape (corresponding angles equal, sides in proportion). Congruent triangles have the same shape and size (corresponding sides and angles equal). Transformations like reflection, rotation, and dilation: reflection and rotation are rigid (preserve shape and size - congruence), dilation changes size (if scale factor ≠ 1, not congruent, but can preserve similarity if angles remain equal).

Step2: Analyze First Statement ($\triangle ABC\sim\triangle A''B''C''$)

Assume no non - similar - changing transformations (if dilation is with a scale factor, but angles remain equal). Since transformations (if any dilation is uniform) preserve angle measures (reflection, rotation preserve angles, dilation preserves angle measures). So, $\triangle ABC$ and $\triangle A''B''C''$ are similar. True.

Step3: Analyze Second Statement ($\triangle ABC\cong\triangle A''B''C''$)

If there is a dilation (scale factor ≠ 1), side lengths change. So, they are not congruent. False.

Step4: Analyze Third Statement (Reflection across y - axis then 90° clockwise rotation)

Reflection across y - axis: $(x,y)\to(-x,y)$. 90° clockwise rotation about origin: $(x,y)\to(y, - x)$. But without seeing exact coordinates (assuming standard grid - based analysis), if the composition of these two rigid transformations (reflection and rotation) can map $\triangle ABC$ to $\triangle A'B'C'$ to $\triangle A''B''C''$ (rigid transformations preserve shape and size). But if there is a dilation in the process (implied by the problem's multiple - transformation nature, but if we consider the statement's transformation sequence: reflection (rigid) and rotation (rigid) - but if the final triangle has different size, this is wrong. Wait, no, the problem says two transformations. Wait, no - the third statement is about a sequence of two transformations (reflection then rotation). But if the actual transformation from $\triangle ABC$ to $\triangle A''B''C''$ involves a dilation (as per the fourth statement's hint of dilation), this sequence (reflection then rotation - both rigid) cannot produce a size - changed triangle. So False.

Step5: Analyze Fourth Statement (Rotation then Dilation)

Rotation (90° counter - clockwise, rigid - preserves shape and size) then dilation (scale factor 2, changes size). If $\triangle ABC$ is rotated (rigid) to $\triangle A'B'C'$ (congruent to $\triangle ABC$) then dilated (scale factor 2) to $\triangle A''B''C''$. Similarity is preserved (angles equal, sides in proportion 1:2). But the problem is about two transformations to get from $\triangle ABC$ to $\triangle A''B''C''$. If the two transformations are rotation (rigid) and dilation (non - rigid, scale factor 2). But we need to check if this sequence is correct. Assuming grid - based side - length analysis (if original side lengths and after - transformation side lengths fit rotation then dilation). True (if we assume the transformation sequence is rotation (rigid) then dilation (scale factor 2)).

Answer:

• $\triangle ABC$ is similar to $\triangle A''B''C''$: True
• $\triangle ABC$ is congruent to $\triangle A''B''C''$: False
• $\triangle ABC$ can be reflected across the $y$ - axis to form $\triangle A'B'C'$, and then $\triangle A'B'C'$ can be rotated $90^{\circ}$ clockwise about the origin to form $\triangle A''B''C''$: False
• $\triangle ABC$ can be rotated $90^{\circ}$ counterclockwise about the origin to form $\triangle A'B'C'$, and then $\triangle A'B'C'$ can be dilated by a scale factor of 2 with the center of dilation at the origin: True