Question

determine whether the statement is always, sometimes, or never true. justify your argument.
a composition of two translations is a rotation.
select choice ; a composition of two translations is select choice select choice

Explanation:

Step1: Recall properties of translations and rotations

Translation moves every point of a figure or space by the same distance in a given direction. Rotation turns a figure around a fixed point (the center of rotation).

Step2: Analyze composition of two translations

Let \(T_1\) be a translation \((x,y)\to(x + a,y + b)\) and \(T_2\) be a translation \((x,y)\to(x + c,y + d)\). The composition \(T_2\circ T_1(x,y)=T_2(T_1(x,y))=(x + a + c,y + b + d)\), which is also a translation (since it moves every point by \((a + c,b + d)\) in a fixed direction). Rotation changes the orientation of a figure (except for rotation by \(360^{\circ}n,n\in\mathbb{Z}\)), while translation preserves the orientation of the figure.

Answer:

Never true. A composition of two translations is a translation (not a rotation as rotation changes orientation in non - \(360^{\circ}n\) cases and translation has a different geometric effect of just shifting the figure).