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: Analyze Option A

Rotation, dilation and translation. But check the scale factor. If \(ABCD\) is dilated by scale factor \(2\), the size relation is wrong. \(ABCD\) is larger than \(PQRS\) in wrong proportion.

Step2: Analyze Option B

Dilation with scale factor \(\frac{1}{2}\) (reduce size), reflection over \(x -\)axis (flip vertically) and translation. Assume coordinates of \(EFGH\) and \(PQRS\). If \(EFGH\) has vertices \((x,y)\), after dilation \((\frac{1}{2}x,\frac{1}{2}y)\), reflection \((\frac{1}{2}x,-\frac{1}{2}y)\) and translation \((\frac{1}{2}x - 4,-\frac{1}{2}y)\). Check similarity (corresponding angles equal and side - length ratios equal).

Step3: Analyze Option C

Translation, dilation and rotation. The order and scale factor combination would not result in similarity as the side - length ratios and angle congruence (due to wrong transformation sequence) would be incorrect.

Step4: Analyze Option D

Rotation and dilation. The scale factor and rotation combination would not map \(PQRS\) to \(ABCD\) correctly in terms of similarity (side - length ratios and orientation after rotation for similarity).

Answer:

B. Trapezoid \(EFGH\) can be mapped onto \(PQRS\) by a dilation with a scale factor of \(\frac{1}{2}\) and center at the origin, followed by a reflection over the \(x -\)axis, then a translation \(4\) units to the left.