Question

Based on the shortest leg of the triangle illustrated, if a similar triangle on the coordinate plane has its shortest leg defined by the points (2, 3) and (8, 3), what is the third point

Answer 1

We can't tell. Contrary to what you led us to believe, there is no triangle illustrated, and we need one for the new one to be similar to.

Answer 2

the complete question in the attached figure

we know that

From the given triangle, we can see that the lengths of the arms of the right-angle are $2$ units and
$3$
units. (the shortest side is
$2 units)$

The second triangle is described as being similar, which means its sides are in the same ratio as the first, but the orientation can be different.

From the two given points the shortest side is horizontal and
the length is equal to

$(8-2)=6 units$

This means that the sides of the second triangle are all
$3$
times as long as the first

$scale factor=3$

The second side of the second triangle will be

$3*3=9 units$

There are $4$ possible positions for the third point.
$9$ units up or down from $(2,3)$ or
$(8,3)$
to form a right-angle.

The options are therefore:

$(2,12)\\ (2,-6)\\ (8,12)\\ (8,-6)$

Only the point
$(2,-6)$ is one of the options given

therefore

the answer is

the third point is
$(2,-6)$