Question

determine whether the statement is always, sometimes, or never true. justify your argument. a composition of two translations is a rotation. never ; a composition of two translations i select choice always sometimes never 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) by a certain angle.
A composition of two translations \((x,y)\to(x + a,y + b)\) followed by \((x,y)\to(x + c,y + d)\) results in \((x,y)\to(x+(a + c),y+(b + d))\), which is also a translation.
Since a rotation changes the orientation of a figure (except for a \(360^{\circ}\) rotation which is equivalent to the identity transformation, but even then, the way of achieving the transformation - through rotation vs. translation - is different in nature for non - \(360^{\circ}\) cases and the composition of two non - trivial translations is not a rotation)

Answer:

Never; a composition of two translations is a translation (since \((x,y)\to(x + a,y + b)\) followed by \((x,y)\to(x + c,y + d)\) gives \((x,y)\to(x+(a + c),y+(b + d))\)), not a rotation.