Question

determine whether each statement about the triangles is true. choose true or false from each drop-down menu for each statement. △abc is similar to △abc. △abc is congruent to △abc. △abc can be reflected across y-axis to form △abc, and then △abc can be rotated 90° clockwise about the origin to form △abc. △abc can be rotated 90° counterclockwise about the origin to form △abc, and then △abc can be dilated by a scale factor of 2 with the center of dilation at the origin to form △abc.

Explanation:

To solve this, we analyze each statement about triangle transformations (similarity, congruence, reflections, rotations, dilations):

1. $\boldsymbol{\triangle ABC}$ is similar to $\boldsymbol{\triangle A''B''C''}$

• Similarity is preserved under reflections, rotations, and dilations (if scale factor is consistent). If $\triangle A''B''C''$ is a transformation (e.g., rotation + dilation) of $\triangle ABC$, their angles remain equal, so they are similar.
• Answer: True

2. $\boldsymbol{\triangle ABC}$ is congruent to $\boldsymbol{\triangle A''B''C''}$

• Congruence requires identical size and shape (no dilation). If $\triangle A''B''C''$ is formed by a dilation (scale factor ≠ 1), it is not congruent. Even without dilation, if transformations include scaling, congruence fails.
• Answer: False

3. $\boldsymbol{\triangle ABC}$ can be reflected across the $y$-axis to form $\boldsymbol{\triangle A'B'C'}$, and then $\boldsymbol{\triangle A'B'C'}$ can be rotated $90^\circ$ clockwise about the origin to form $\boldsymbol{\triangle A''B''C''}$

• Reflecting over $y$-axis flips the triangle. Rotating $90^\circ$ clockwise transforms it further. However, if the final triangle $\triangle A''B''C''$ does not match the result of these two transformations (e.g., due to scaling or incorrect rotation), this is false. Typically, such a sequence (reflection + rotation) does not produce the target triangle if dilation is involved or the rotation is misaligned.
• Answer: False

4. $\boldsymbol{\triangle ABC}$ can be rotated $90^\circ$ counterclockwise about the origin to form $\boldsymbol{\triangle A'B'C'}$, and then $\boldsymbol{\triangle A'B'C'}$ can be dilated by a scale factor of 2 with the center of dilation at the origin to form $\boldsymbol{\triangle A''B''C''}$

• Rotation preserves shape/size; dilation by 2 scales it. If $\triangle A''B''C''$ is twice the size of $\triangle ABC$ (after rotation), this sequence works.
• Answer: True

Final Answers (for each dropdown):

1. $\triangle ABC$ is similar to $\triangle A''B''C''$: $\boldsymbol{\text{True}}$
2. $\triangle ABC$ is congruent to $\triangle A''B''C''$: $\boldsymbol{\text{False}}$
3. Reflection + $90^\circ$ clockwise rotation: $\boldsymbol{\text{False}}$
4. $90^\circ$ counterclockwise rotation + dilation (scale 2): $\boldsymbol{\text{True}}$

Answer:

To solve this, we analyze each statement about triangle transformations (similarity, congruence, reflections, rotations, dilations):

1. $\boldsymbol{\triangle ABC}$ is similar to $\boldsymbol{\triangle A''B''C''}$

• Similarity is preserved under reflections, rotations, and dilations (if scale factor is consistent). If $\triangle A''B''C''$ is a transformation (e.g., rotation + dilation) of $\triangle ABC$, their angles remain equal, so they are similar.
• Answer: True

2. $\boldsymbol{\triangle ABC}$ is congruent to $\boldsymbol{\triangle A''B''C''}$

• Congruence requires identical size and shape (no dilation). If $\triangle A''B''C''$ is formed by a dilation (scale factor ≠ 1), it is not congruent. Even without dilation, if transformations include scaling, congruence fails.
• Answer: False

3. $\boldsymbol{\triangle ABC}$ can be reflected across the $y$-axis to form $\boldsymbol{\triangle A'B'C'}$, and then $\boldsymbol{\triangle A'B'C'}$ can be rotated $90^\circ$ clockwise about the origin to form $\boldsymbol{\triangle A''B''C''}$

• Reflecting over $y$-axis flips the triangle. Rotating $90^\circ$ clockwise transforms it further. However, if the final triangle $\triangle A''B''C''$ does not match the result of these two transformations (e.g., due to scaling or incorrect rotation), this is false. Typically, such a sequence (reflection + rotation) does not produce the target triangle if dilation is involved or the rotation is misaligned.
• Answer: False

4. $\boldsymbol{\triangle ABC}$ can be rotated $90^\circ$ counterclockwise about the origin to form $\boldsymbol{\triangle A'B'C'}$, and then $\boldsymbol{\triangle A'B'C'}$ can be dilated by a scale factor of 2 with the center of dilation at the origin to form $\boldsymbol{\triangle A''B''C''}$

• Rotation preserves shape/size; dilation by 2 scales it. If $\triangle A''B''C''$ is twice the size of $\triangle ABC$ (after rotation), this sequence works.
• Answer: True

Final Answers (for each dropdown):

1. $\triangle ABC$ is similar to $\triangle A''B''C''$: $\boldsymbol{\text{True}}$
2. $\triangle ABC$ is congruent to $\triangle A''B''C''$: $\boldsymbol{\text{False}}$
3. Reflection + $90^\circ$ clockwise rotation: $\boldsymbol{\text{False}}$
4. $90^\circ$ counterclockwise rotation + dilation (scale 2): $\boldsymbol{\text{True}}$