Question

a triangle in quadrant i is reflected over the y - axis, then over the x - axis, and then over the y - axis again. the new triangle, in quadrant iv, could also have been obtained by using a single

Explanation:

Step1: Understand reflection properties

Reflecting a point \((x,y)\) over the \(y -\)axis gives \((-x,y)\), then reflecting \((-x,y)\) over the \(x -\)axis gives \((-x,-y)\), and then reflecting \((-x,-y)\) over the \(y -\)axis gives \((x,-y)\).
A rotation of \(180^{\circ}\) about the origin takes a point \((x,y)\) to \((-x,-y)\), and then a reflection over the \(x -\)axis takes \((-x,-y)\) to \((-x,y)\) (incorrect). A dilation changes the size (not relevant here). A translation slides the figure (not relevant as orientation changes).
A single reflection over the \(x -\)axis takes \((x,y)\) to \((x,-y)\) which is the same as the composition of \(y -\)axis reflection, \(x -\)axis reflection and \(y -\)axis reflection.

Answer:

B. reflection.