Question

select all transformations or compositions of transformations that map the figure onto itself.
reflection in the line ( y = 2.5 )
reflection in the line ( x = 1 )
reflection in the line ( x = - 1 ) followed by the translation ( (x,y)\to(x + 4,y) )
rotation ( 180^{circ} ) about ( (-1,1) ) followed by a reflection in the line ( y = 1 )
reflection in the line ( y = 4 ) followed by the translation ( (x,y)\to(x,y - 3) )

Explanation:

Step1: Analyze reflection in \(y = 2.5\)

The line \(y=2.5\) is the horizontal line that is the perpendicular bisector of the vertical segments connecting the top - and bottom - horizontal sides of the trapezoid. Reflecting the trapezoid over \(y = 2.5\) maps the trapezoid onto itself.

Step2: Analyze reflection in \(x = 1\)

The line \(x = 1\) is not a line of symmetry for the trapezoid. If we take a point \((x,y)\) on the trapezoid and reflect it over \(x=1\) (using the formula \((x,y)\to(2 - x,y)\)), the reflected figure will not coincide with the original trapezoid.

Step3: Analyze reflection in \(x=-1\) followed by translation \((x,y)\to(x + 4,y)\)

First, reflection over \(x=-1\) (using the formula \((x,y)\to(-2 - x,y)\)). Then, translation \((x,y)\to(x + 4,y)\) (i.e., \((-2 - x,y)\to(-2 - x+4,y)=(2 - x,y)\)). This is equivalent to a reflection over \(x = 1\) (which we saw is not a line of symmetry), so this composition does not map the trapezoid onto itself.

Step4: Analyze rotation \(180^{\circ}\) about \((-1,1)\) followed by reflection in \(y = 1\)

A \(180^{\circ}\) rotation about \((-1,1)\) (using the formula \((x,y)\to(-2 - x,2 - y)\)) followed by reflection over \(y = 1\) (using the formula \((x,y)\to(x,2 - y)\)). The composition \((x,y)\to(-2 - x,2 - y)\to(-2 - x,2-(2 - y))=(-2 - x,y)\) does not map the trapezoid onto itself.

Step5: Analyze reflection in \(y = 4\) followed by translation \((x,y)\to(x,y - 3)\)

Reflection over \(y = 4\) (using the formula \((x,y)\to(x,8 - y)\)) followed by translation \((x,y)\to(x,y - 3)\) (i.e., \((x,8 - y)\to(x,8 - y-3)=(x,5 - y)\)). The top - horizontal side of the trapezoid is at \(y = 4\) and the bottom - horizontal side is at \(y=1\). If we take a point \((x,4)\) (on the top - side) and apply the transformation: \((x,4)\to(x,8 - 4)=(x,4)\to(x,4 - 3)=(x,1)\) and vice - versa. This composition maps the trapezoid onto itself.

Answer:

reflection in the line \(y = 2.5\), reflection in the line \(y = 4\) followed by the translation \((x,y)\to(x,y - 3)\)