Question

question 4 (multiple - choice worth 1 points) (02.03 mc) what series of transformations would carry the trapezoid onto itself? (x + 0,y - 4), 90° clockwise rotation, reflection over the y - axis (x + 0,y - 4), 180° rotation, reflection over the y - axis (x + 6,y + 0), 90° clockwise rotation, reflection over the x - axis (x + 6,y + 0), 180° rotation, reflection over the x - axis

Explanation:

Step1: Analyze translation effect

For a translation $(x + a,y + b)$, it moves the figure. $(x+0,y - 4)$ moves the trapezoid 4 units down. $(x + 6,y+0)$ moves it 6 units right.

Step2: Analyze rotation effect

A $90^{\circ}$ clock - wise rotation about the origin changes the orientation of the figure. The rule for a $90^{\circ}$ clock - wise rotation of a point $(x,y)$ is $(y,-x)$. A $180^{\circ}$ rotation about the origin of a point $(x,y)$ is $(-x,-y)$.

Step3: Analyze reflection effect

Reflection over the $x$ - axis changes the sign of the $y$ - coordinate $(x,y)\to(x, - y)$. Reflection over the $y$ - axis changes the sign of the $x$ - coordinate $(x,y)\to(-x,y)$.

Step4: Check each option

Let's assume the trapezoid has vertices. For the first option: $(x + 0,y - 4)$ moves it down, $90^{\circ}$ clock - wise rotation changes its orientation and reflection over $y$ - axis changes it further, not mapping it onto itself. For the second option: $(x + 0,y - 4)$ moves it down, $180^{\circ}$ rotation and reflection over $y$ - axis do not map it onto itself. For the third option: $(x + 6,y+0)$ moves it right, $90^{\circ}$ clock - wise rotation and reflection over $x$ - axis do not map it onto itself. For the fourth option: Let's consider a trapezoid's properties. A translation $(x + 6,y + 0)$ moves the trapezoid 6 units right. A $180^{\circ}$ rotation about the origin and then a reflection over the $x$ - axis can map the trapezoid onto itself if it has certain symmetries.

Answer:

$(x + 6,y + 0),180^{\circ}$ rotation, reflection over the $x$ - axis