Video Transcript
The image of point 𝑋 is 𝑋
prime following a translation of magnitude 𝑀𝑁 in the direction of the ray
𝑀𝑁. Which of the following diagrams
represents this?
Starting with the point in
space 𝑋, we are told that the translation is of magnitude 𝑀𝑁. This means that the image of
𝑋, 𝑋 prime, will lie somewhere on the circle of radius 𝑀𝑁 centered at the
original point 𝑋. We also have the ray 𝑀𝑁,
which points in the direction of the translation. We can therefore construct a
ray starting from the point 𝑋 that is parallel and in the same direction to the
ray 𝑀𝑁. And where this new ray touches
the circle will be where the image 𝑋 prime lies.
To do this, we start by
constructing a ray that passes through 𝑋 in the direction of 𝑀. We seek to duplicate the angle
𝑁𝑀𝑋 at the point 𝑋. We can do this by constructing
two arcs of circles of equal radius centered at the points 𝑋 and 𝑀, labeling
the points of intersection between the circle centered at 𝑀 and the two rays 𝐴
and 𝐵. We then measure this distance
between 𝐴 and 𝐵 and draw another arc of another circle centered on the far
point of intersection between the circle arc centered at 𝑋 and the ray passing
through 𝑋 and 𝑀. A ray passing through the point
𝑋 and the point of intersection of these two arcs will be parallel to the ray
𝑀𝑁.
We can therefore draw the ray
starting at the point 𝑋 and passing through this point of intersection. And the point of intersection
between this ray and the circle will be the image of the point 𝑋 prime. Looking at the possible
answers, we can see that this matches with the diagram in 𝐴.
