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△EFG is rotated −90 degrees (clockwise) to create △XYZ . Which of the following statements must be true?(1 point) Responses

3 answers

To answer your question regarding the rotation of triangle \( \Delta EFG \) by -90 degrees (clockwise) to create triangle \( \Delta XYZ \), we should consider the properties of the transformation and the relationships between the points.

When a triangle is rotated -90 degrees clockwise around a point (typically the origin unless stated otherwise), each point of the triangle is moved according to the following transformation:

For a point \( (x, y) \), the new coordinates after a -90-degree rotation will be \( (y, -x) \).

Therefore:

  • Point \( E \) will rotate to a new position \( X \).
  • Point \( F \) will rotate to a new position \( Y \).
  • Point \( G \) will rotate to a new position \( Z \).

Here are some statements that can be true regarding the triangles after the rotation:

  1. Angles: The angles of \( \Delta XYZ \) will be equal to the corresponding angles of \( \Delta EFG \). Thus, if \( \angle EFG = \angle XYZ \), this is a true statement.

  2. Congruence: The two triangles \( \Delta EFG \) and \( \Delta XYZ \) are congruent, meaning that they have the same size and shape.

  3. Position: The position and orientation of triangle \( \Delta XYZ \) will be different compared to triangle \( \Delta EFG \) due to the rotation.

  4. Length of Sides: The lengths of the sides of both triangles will be equal (i.e., \( EF = XY \), \( FG = YZ \), and \( GE = ZX \)).

Based on the properties of rotations, statements regarding congruence, length of sides, and angle equality must be true following a rotation transformation.

If you'd like specific statements to consider, please provide them, and I can confirm which ones are true.

△EFG is rotated −90 degrees (clockwise) to create △XYZ . Which of the following statements must be true?(1 point) Responses ∠E≅∠X angle upper E congruent to angle upper X EF¯¯¯¯¯¯¯¯≅XZ¯¯¯¯¯¯¯¯ Modifying above upper E upper F with bar congruent to Modifying above upper X upper Z with bar ∠F≅∠Z angle upper F congruent to angle upper Z EG¯¯¯¯¯¯¯¯≅XY¯¯¯¯¯¯¯¯

To determine which statements must be true after the rotation of triangle \( \Delta EFG \) by -90 degrees clockwise to create triangle \( \Delta XYZ \), let's analyze the options provided:

Given Statements:

  1. \( \angle E \cong \angle X \) (Angle E is congruent to angle X)
  2. \( EF \cong XZ \) (Segment EF is congruent to segment XZ)
  3. \( \angle F \cong \angle Z \) (Angle F is congruent to angle Z)
  4. \( EG \cong XY \) (Segment EG is congruent to segment XY)

Analysis:

  • Angle Congruence: When a triangle is rotated, the angles remain congruent. Therefore:

    • \( \angle E \cong \angle X \) should be true. (True)
    • \( \angle F \cong \angle Z \) should also be true. (True)

  • Side Length Congruence: The lengths of corresponding sides also remain congruent when a triangle is rotated. Therefore:

    • \( EF \cong XZ \) should be true. (False, because after a -90-degree rotation, the segments don't correspond this way.)
    • \( EG \cong XY \) should also be false based on the rotation. (False, because after a -90-degree rotation, the segments don't match this way.)

Correct Responses:

  1. \( \angle E \cong \angle X \) (True)
  2. \( EF \cong XZ \) (Not True)
  3. \( \angle F \cong \angle Z \) (True)
  4. \( EG \cong XY \) (Not True)

Conclusion:

The statements that must be true are:

  • \( \angle E \cong \angle X \)
  • \( \angle F \cong \angle Z \)

The statements about the side lengths (EF and XZ, EG and XY) are not true based on their positions after rotation. Thus, only the first and third statements are guaranteed to be true after the transformation.