Question

which transformation(s) must map the isosceles trapezoid exactly onto itself? choose all that apply. counterclockwise rotation about y by 180° reflection across line b clockwise rotation about y by 90° reflection across line a none of the above

Explanation:

Step1: Analyze rotation about \( Y \)

• A counter - clockwise rotation about \( Y \) by \( 180^{\circ} \):
• For an isosceles trapezoid, when rotated \( 180^{\circ} \) about the mid - point \( Y \) (the intersection of the axes of symmetry in some sense), each point \( (x,y) \) is mapped to \( (-x,-y) \). Since the trapezoid is symmetric with respect to the center \( Y \) (in terms of the property that the non - parallel sides are equal and the bases are parallel), this rotation maps the trapezoid onto itself.
• A clockwise rotation about \( Y \) by \( 90^{\circ} \):
• An isosceles trapezoid does not have rotational symmetry of order \( 4 \) (a \( 90^{\circ} \) rotation). The shape of the isosceles trapezoid (with only one pair of parallel sides and non - parallel sides equal) will not map onto itself after a \( 90^{\circ} \) rotation.

Step2: Analyze reflection

• Reflection across line \( b \):
• Line \( b \) is a vertical line passing through the mid - point \( Y \) of the non - parallel sides (assuming the standard properties of an isosceles trapezoid drawn with a vertical axis of symmetry). A reflection across a vertical line of symmetry of an isosceles trapezoid maps the trapezoid onto itself.
• Reflection across line \( a \):
• Line \( a \) is not an axis of symmetry of the isosceles trapezoid. An isosceles trapezoid has only one vertical axis of symmetry (in the standard case, if we assume the bases are horizontal). A reflection across a non - axis of symmetry line will not map the trapezoid onto itself.

Answer:

Counterclockwise rotation about \( Y \) by \( 180^{\circ} \), Reflection across line \( b \)