At the bottom, try to define in your own words what a translation is.

A [color=#0000ff]translation slides a figure[/color] in a given direction for a given distance with no rotation. The distance and direction is given by a [b]directed line segment[/b]. The arrow of the directed line segment specifies the [i][b]direction [/b][/i]of the translation, and the[i] [b]length[/b][/i] of the directed line segment specifies how far the figure gets translated. [br][img]https://cms-im.s3.amazonaws.com/PpUBt1WJrZDnCDjQxiw6tc31?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-6402.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-6402.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJFC4WL6K24MWHIRQ%2F20211006%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20211006T015802Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=a0289f62715a450babb0b23c98992ab2c5850dbfff6d73566df0722d28952833[/img]

A [b]translation[/b] of a point [math]A[/math] along a [i]directed line segment[/i] [math]t[/math] is [color=#0000ff]a transformation that takes [/color][math]A[/math][color=#0000ff] to [/color][math]A'[/math] so that the [color=#0000ff]directed line segment[/color] [math]AA'[/math] is parallel to [math]t[/math], [color=#0000ff]goes in the same direction as [/color][math]t[/math][color=#0000ff], and is the same length as [/color][math]t[/math][color=#0000ff]. [/color][br][img]https://cms-im.s3.amazonaws.com/8TLNj6g7jgKH6W4vF4DMVY7h?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-6403.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-6403.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJFC4WL6K24MWHIRQ%2F20211006%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20211006T015802Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=f88b3b4c3ab8b8af2870631d0f6bc15d9b0c9f6dd540753ce6bc27c80a1a6cbb[/img]

Here is a translation of 3 points. Notice that the directed line segments [math]CC'[/math], [math]DD'[/math], and [math]EE'[/math] are each parallel to [math]v[/math], going in the same direction as [math]v[/math], and the same length as [math]v[/math]. [br][img]https://cms-im.s3.amazonaws.com/1r81R4TTKHhph3cYfKy6GjUo?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-6404.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-6404.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJFC4WL6K24MWHIRQ%2F20211006%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20211006T015802Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=775ecf35bd022b629b6b038592a68bb22e63fd7ec74cb82b2de064caa3479608[/img]