Question

which of the following is true for the following transformation.
(x,y)--> (-x,y)
the orientation of vertices is preserved.
the area is the same.
the side lengths are congruent.
the angle measures are congruent.

Explanation:

The transformation \((x,y)\to(-x,y)\) is a reflection over the \(y\) - axis.

• Orientation of vertices: A reflection changes the orientation of the vertices (for example, a clock - wise oriented figure will become counter - clockwise after reflection over the \(y\) - axis), so the statement "The orientation of vertices is preserved" is false.
• Area: Reflections are rigid transformations. Rigid transformations (reflections, translations, rotations) do not change the area of a figure. So the area of the figure before and after the reflection will be the same.
• Side lengths: Since reflections are rigid transformations, the distance between any two points (which determines the side length of a polygon) is preserved. If we have two points \((x_1,y_1)\) and \((x_2,y_2)\) before reflection, and their images \((-x_1,y_1)\) and \((-x_2,y_2)\) after reflection, the distance \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) and the distance between the images \(d'=\sqrt{(-x_2-(-x_1))^2+(y_2 - y_1)^2}=\sqrt{(x_1 - x_2)^2+(y_2 - y_1)^2}=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}=d\). So side lengths are congruent.
• Angle measures: Rigid transformations also preserve angle measures. When we reflect a figure, the angles between the sides of the figure remain the same because the relative positions of the sides (in terms of their slopes and the way they intersect) are preserved up to the reflection, and the angle between two lines is determined by their slopes (or the vectors representing the sides), and reflection over the \(y\) - axis does not change the angle between two lines.

Answer:

The area is the same.
The side lengths are congruent.
The angle measures are congruent.