Question

1. select the composition of transformations that would not map trapezoid qrst onto trapezoid defg.

a a reflection over the x - axis, then a reflection over x = - 1.
b a rotation of 180° about the origin, followed by a translation two units to the left
c a reflection over the y - axis, then a translation two units to the left and eight uni
d a translation two units to the right, followed by a rotation of 180° about the ori
a dilation with a scale factor of 1.

Explanation:

Step1: Analyze Option A

• Reflect trapezoid \(QRST\) over the \(x -\)axis. Then reflect the resulting figure over \(x=-1\).
• The first reflection (over \(x -\)axis) changes the \(y -\)coordinates of the vertices of \(QRST\) (e.g., if a point \((x,y)\) on \(QRST\), it becomes \((x, - y)\) after reflection over \(x -\)axis). The second reflection (over \(x =-1\)) uses the formula \((x,y)\to(-2 - x,y)\). This composition of reflections is a rigid - motion (isometry) and can map \(QRST\) onto \(DEFG\).

Step2: Analyze Option B

• Rotate trapezoid \(QRST\) \(180^{\circ}\) about the origin. The rule for a \(180^{\circ}\) rotation about the origin is \((x,y)\to(-x,-y)\). Then translate the resulting figure \(2\) units to the left (using the rule \((x,y)\to(x - 2,y)\)).
• A \(180^{\circ}\) rotation about the origin is a rigid - motion. Translation is also a rigid - motion. The composition of these two transformations (rotation and translation) is a rigid - motion (isometry) and can map \(QRST\) onto \(DEFG\).

Step3: Analyze Option C

• Reflect trapezoid \(QRST\) over the \(y -\)axis (rule \((x,y)\to(-x,y)\)). Then translate the resulting figure \(2\) units to the left (rule \((x,y)\to(x - 2,y)\)) and \(8\) units up (rule \((x,y)\to(x,y + 8)\)).
• Reflection over \(y -\)axis is a rigid - motion. Translation is a rigid - motion. The composition of reflection and translation is a rigid - motion (isometry) and can map \(QRST\) onto \(DEFG\).

Step4: Analyze Option D

• Translate trapezoid \(QRST\) \(2\) units to the right (rule \((x,y)\to(x + 2,y)\)). Then rotate \(180^{\circ}\) about the origin (rule \((x,y)\to(-x,-y)\)). A dilation with a scale factor of \(1\) (rule \((x,y)\to(x,y)\)) does nothing.
• A \(180^{\circ}\) rotation about the origin changes the orientation of the trapezoid. After translation \(2\) units to the right and \(180^{\circ}\) rotation about the origin, the orientation of the trapezoid \(QRST\) (with the non - parallel sides) does not match the orientation of \(DEFG\) (the non - parallel sides are in a different relative position).

Answer:

D. A translation two units to the right, followed by a rotation of \(180^{\circ}\) about the origin and a dilation with a scale factor of \(1\).