Question

determine whether the statement is always, sometimes, or never true. justify your argument.
a composition of two translations is a rotation.
select choice
always
sometimes
never
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).
Let the first translation be \(T_1(x,y)=(x + a,y + b)\) and the second translation be \(T_2(x,y)=(x + c,y + d)\).
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.
A rotation \((x,y)\to(x\cos\theta - y\sin\theta,x\sin\theta + y\cos\theta)\) (for non - zero \(\theta\)) changes the orientation of the figure in a non - translational way.

Answer:

Never. A composition of two translations is a translation (since \(T_2\circ T_1(x,y)=(x+(a + c),y+(b + d))\)), and a translation is not a rotation (rotation changes orientation around a point in a non - translational distance - preserving but direction - changing way, while translation just shifts the figure).