Video Transcript
Which graph represents
reflecting the triangle 𝐴𝐵𝐶 about the origin?
We’re asked about the effect of
reflecting triangle 𝐴𝐵𝐶 in a point, which is the origin. So triangle 𝐴𝐵𝐶 is the
object, and we need to determine which of the four triangles is the correct
image for this transformation.
We recall that reflection about
the origin maps a general point 𝑃 with coordinates 𝑥, 𝑦 to the point 𝑃 prime
with coordinates negative 𝑥, negative 𝑦. Or in other words, 𝑃 and 𝑃
prime have the same coordinates but with opposite signs. The two points are the same
distance from the origin, but on opposite sides of it in a straight line.
We can see immediately then
that graph (A) cannot be the correct graph, because whilst the 𝑥-coordinates
have changed sign, the 𝑦-coordinates have not. Also, if we draw straight lines
from each vertex of triangle 𝐴𝐵𝐶 to be the origin and continue these lines,
we can see that the image of triangle 𝐴𝐵𝐶 after reflection about the origin
should be in the third quadrant. For this reason, we can
actually eliminate both (A) and (B) because the image is in the incorrect
quadrant. In graphs (C) and (D), however,
both images are in the third quadrant.
We can now recall a key
property of reflection in the origin, which is that orientation of the vertices
is preserved. This means that if the vertices
of the object are labeled in, for example, the clockwise direction, the same
will be true for the vertices of the image. Looking at triangle 𝐴𝐵𝐶, we
can see that the ordering of the vertices is in fact counterclockwise, so the
same must be true for its image.
On graph (C), the direction of
labeling is also counterclockwise. But on graph (D), the direction
is clockwise. This allows us to rule option
(D) out. Only graph (C) remains. And we can check that this is
indeed correct by connecting corresponding vertices and observing that they are
indeed equal distance from the origin but on opposite sides in a straight
line. So option (C) is the correct
graph to represent a reflection of triangle 𝐴𝐵𝐶 about the origin.
