Question

1. in a new translation, the image of v is z. find at least 3 other points that are taken to a labeled point by the new translation.
2. are your conjectures still true for the new translation?

Explanation:

Step1: Recall translation property

A translation is a rigid - motion that moves every point of a figure the same distance in the same direction. If point $V$ is translated to $Z$, we can find other points by observing the vector from $V$ to $Z$.

Step2: Assume we have a coordinate - like or grid - based figure

Let's assume we can measure the horizontal and vertical displacements from $V$ to $Z$. If the displacement from $V$ to $Z$ is $(x,y)$ units (horizontal and vertical respectively), we can find other points. For example, if there is a point $A$ near $V$, we can find its image $A'$ by applying the same displacement $(x,y)$.

Step3: Without loss of generality

Suppose we have points $W$, $X$, $Y$ near $V$. If we apply the translation vector that takes $V$ to $Z$ to $W$, $X$, $Y$, we get their images $W'$, $X'$, $Y'$ which may be labeled points.

Answer:

Let's assume we have points $A$, $B$, $C$ near $V$. After applying the translation that takes $V$ to $Z$, the images of $A$, $B$, $C$ (say $A'$, $B'$, $C'$) may be labeled points. For the second question, if the conjectures are related to properties preserved under translation (such as distance - preservation, angle - preservation), then yes, the conjectures are still true for the new translation since translation is a rigid motion and preserves distances and angles.