Question

△abc can be reflected across the y-axis to form △a′b′c′, and then △a′b′c′ can be rotated 90° clockw about the origin to form △a′′b′′c′′ △abc can be rotated 90° counterclockwise about the origir form △a′b′c′, and then △a′b′c′ c

Explanation:

To determine the answer, we analyze the transformations:

For the first statement:

Reflecting a triangle over the \( y \)-axis and then rotating \( 90^\circ \) clockwise about the origin is a valid sequence of rigid transformations (reflection and rotation preserve shape and size). However, the original answer marked it as "False," which is incorrect. Wait, no—wait, the problem’s context (likely missing coordinates of \( \triangle ABC \)) is needed. But assuming standard transformation rules: A reflection over \( y \)-axis changes \( (x,y) \to (-x,y) \), and a \( 90^\circ \) clockwise rotation about origin changes \( (x,y) \to (y,-x) \). Combining these: \( (-x,y) \to (y, x) \). If the final triangle \( \triangle A''B''C'' \) matches the target, the statement could be True. But the original dropdown had "False" selected, which may be incorrect. Wait, the second statement: Rotating \( 90^\circ \) counterclockwise about origin (rule \( (x,y) \to (-y,x) \)) and then... (text cut off). But the checked "False" for the second statement suggests it’s invalid.

Correcting (assuming the first statement’s validity):

Wait, the first statement’s dropdown is "Choose..."—let’s re-express. If the question is whether the sequence (reflect over \( y \)-axis, then rotate \( 90^\circ \) clockwise) works, the answer depends on the triangle’s coordinates. But if we assume the standard, the first statement is True (since both are rigid transformations, and sequences of rigid transformations map triangles to congruent triangles). The second statement (rotating \( 90^\circ \) counterclockwise then... )—if the "then" part is, say, reflecting, but the text is cut off. However, the checked "False" for the second statement implies it’s invalid.

For the first "Choose..." dropdown (the top one):
If the transformation sequence is valid (reflection + rotation), the answer is True.

For the second (with "√ False"):
If the rotation + subsequent transformation is invalid, "False" is correct.

But since the first dropdown is "Choose...", let's assume the intended answer for the first statement (after reflection and rotation) is True (as rigid transformations preserve congruence, and the sequence is valid).

(for the first "Choose..." dropdown):
Rigid transformations (reflection, rotation) preserve triangle congruence. Reflecting over \( y \)-axis and rotating \( 90^\circ \) clockwise is a valid sequence to map \( \triangle ABC \) to \( \triangle A''B''C'' \) (if coordinates align). Thus, the statement is True.

Answer:

(for the first "Choose..." dropdown):
True