Question

1. the graph shows trapezoids abcd, efgh, and pqrs. select the statement that explains why a pair of trapezoids are similar to each other.

a trapezoid abcd can be mapped onto pqrs by a clockwise rotation of 90° with the center at the origin, followed by a dilation with a scale factor of 2 and center at the origin, and finally a translation 4 units to the left and 4 units down.
b trapezoid efgh can be mapped onto pqrs by a dilation with a scale factor of 1/2 and center at the origin, followed by a reflection over the x - axis, then a translation 4 units to the left.
c trapezoid pqrs can be mapped onto abcd by a translation 4 units to the right and 4 units up, followed by a dilation with a scale factor of 2 with the center at the origin, and finally a 270° clockwise rotation about the origin.
d trapezoid pqrs can be mapped onto abcd by a clockwise rotation of 90°, followed by a dilation with the scale factor of 2 and center at the origin.

Explanation:

Step1: Recall Similarity Transformations

Similarity transformations include rotations, dilations, translations, and reflections. For two figures to be similar, one can be mapped to the other via a combination of these, with dilation (scale factor ≠ 0) and rigid motions (rotation, translation, reflection).

Step2: Analyze Option D

• Rotation: A clockwise rotation of \(90^\circ\) about the origin. This preserves shape and angles (rigid motion).
• Dilation: Scale factor 2, center origin. This scales the figure, preserving similarity (angles same, sides proportional).

If \(PQRS\) is rotated \(90^\circ\) clockwise and then dilated by 2, it should map to \(ABCD\) (checking orientation, side ratios, and position). Other options have errors:

• Option A: Incorrect translation/dilation order or direction.
• Option B: Reflection over x - axis is unnecessary and scale factor \(1/2\) would shrink, not match \(ABCD\) from \(EFGH\).
• Option C: 270° rotation and translation/dilation order is incorrect.

Answer:

D. Trapezoid \(PQRS\) can be mapped onto \(ABCD\) by a clockwise rotation of \(90^\circ\), followed by a dilation with the scale factor of 2 and center at the origin.