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. true △abc is congruent to △abc. choose... △abc can be reflected across the y-axis to form △abc, and then △abc can be rotated 90° clockwise about the origin to form △abc. choose... △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. choose...

Explanation:

To solve this, we analyze each statement using triangle congruence, similarity, and transformation rules:

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

Congruent triangles have identical size and shape (same side lengths, angles). If $\triangle A''B''C''$ is a dilation (scaling) or different size, they are not congruent. Assume $\triangle A''B''C''$ is a scaled version (e.g., from dilation) → False.

2. $\boldsymbol{\triangle ABC \text{ reflected over } y\text{-axis, then rotated } 90^\circ \text{ clockwise to form } \triangle A''B''C''}$

• Reflection over $y$-axis: $(x,y) \to (-x,y)$.
• Rotation $90^\circ$ clockwise: $(-x,y) \to (y,x)$.

If the final coordinates match $\triangle A''B''C''$, this is true. If not (e.g., dilation is involved), it’s false. Assume the transformations are rigid (no scaling) → True (if steps align) or False (if scaling occurs). For typical problems, this is often False (e.g., if $\triangle A''B''C''$ is scaled).

3. $\boldsymbol{\triangle ABC \text{ rotated } 90^\circ \text{ counterclockwise, then dilated by 2 to form } \triangle A''B''C''}$

• Rotation $90^\circ$ counterclockwise: $(x,y) \to (-y,x)$.
• Dilation by 2: $(-y,x) \to (-2y, 2x)$.

If $\triangle A''B''C''$ has double the side length (scaled), this matches dilation. So the sequence (rotation + dilation) works → True.

Final Answers (assuming standard problem context):

• $\triangle ABC$ is congruent to $\triangle A''B''C''$: $\boldsymbol{\text{False}}$
• $\triangle ABC$ reflected then rotated: $\boldsymbol{\text{False}}$ (if scaling is involved)
• $\triangle ABC$ rotated then dilated: $\boldsymbol{\text{True}}$

(Note: Exact answers depend on the triangles’ coordinates, but the above follows general transformation rules.)

Answer:

To solve this, we analyze each statement using triangle congruence, similarity, and transformation rules:

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

Congruent triangles have identical size and shape (same side lengths, angles). If $\triangle A''B''C''$ is a dilation (scaling) or different size, they are not congruent. Assume $\triangle A''B''C''$ is a scaled version (e.g., from dilation) → False.

2. $\boldsymbol{\triangle ABC \text{ reflected over } y\text{-axis, then rotated } 90^\circ \text{ clockwise to form } \triangle A''B''C''}$

• Reflection over $y$-axis: $(x,y) \to (-x,y)$.
• Rotation $90^\circ$ clockwise: $(-x,y) \to (y,x)$.

If the final coordinates match $\triangle A''B''C''$, this is true. If not (e.g., dilation is involved), it’s false. Assume the transformations are rigid (no scaling) → True (if steps align) or False (if scaling occurs). For typical problems, this is often False (e.g., if $\triangle A''B''C''$ is scaled).

3. $\boldsymbol{\triangle ABC \text{ rotated } 90^\circ \text{ counterclockwise, then dilated by 2 to form } \triangle A''B''C''}$

• Rotation $90^\circ$ counterclockwise: $(x,y) \to (-y,x)$.
• Dilation by 2: $(-y,x) \to (-2y, 2x)$.

If $\triangle A''B''C''$ has double the side length (scaled), this matches dilation. So the sequence (rotation + dilation) works → True.

Final Answers (assuming standard problem context):

• $\triangle ABC$ is congruent to $\triangle A''B''C''$: $\boldsymbol{\text{False}}$
• $\triangle ABC$ reflected then rotated: $\boldsymbol{\text{False}}$ (if scaling is involved)
• $\triangle ABC$ rotated then dilated: $\boldsymbol{\text{True}}$

(Note: Exact answers depend on the triangles’ coordinates, but the above follows general transformation rules.)