The figure shows two triangles
𝐴𝐵𝐶 and 𝐴 prime 𝐵 prime 𝐶 prime. Firstly, describe the single
transformation that would map 𝐴𝐵𝐶 to 𝐴 prime 𝐵 prime 𝐶 prime. Secondly, hence, determine whether
triangles 𝐴𝐵𝐶 and 𝐴 prime 𝐵 prime 𝐶 prime are similar.
Firstly, we’re asked for the single
transformation that maps 𝐴𝐵𝐶 to 𝐴 prime 𝐵 prime 𝐶 prime. This word single is really
important. We can’t give a combination of
transformations as our answer because the question tells us that it’s possible to
achieve this with just one single transformation.
Let’s look at the triangles more
closely. The two triangles have different
sizes, which tells us that the type of transformation we’re looking at must be a
dilation. We can determine the scale factor
of this dilation by looking at the length of corresponding sides in the two
The base of triangle 𝐴𝐵𝐶 is one
unit, whereas the base of triangle 𝐴 prime 𝐵 prime 𝐶 prime is two units. The perpendicular height of 𝐴𝐵𝐶
is two, and the perpendicular height of 𝐴 prime 𝐵 prime 𝐶 prime is four. Therefore, the sides in triangle 𝐴
prime 𝐵 prime 𝐶 prime are always twice as long as the corresponding sides in
triangle 𝐴𝐵𝐶. So the scale factor of the dilation
Finally, let’s determine the
coordinates of the point that this dilation has occurred from. To do this, we need to draw in
lines or rays connecting the corresponding vertices of the two triangles. First we draw in the line
connecting 𝐴 and 𝐴 prime. Next I draw in the line connecting
𝐶 and 𝐶 prime. Now actually, it would be enough
just to draw these two rays, but I’ll draw the third in as well.
What you’ll notice is that these
three rays all intersect at a common point, and it is this point that the dilation
has occurred from. This point has the coordinates
negative three, zero. Therefore, our answer to the first
part of the question, the single transformation that maps 𝐴𝐵𝐶 to 𝐴 prime 𝐵
prime 𝐶 prime is the dilation from the point negative three, zero, with scale
The second part of the question
asked us to determine whether the two triangles are similar. Well, if one triangle is an
enlargement of the other, as is the case with the dilation, then all the
corresponding lengths will be in the same ratio and all the corresponding angles
will be the same size. So yes, the two triangles will be
similar to each other.
So we have our answer to the first
part of the problem; the single transformation is a dilation from the point negative
three, zero, with scale factor two. And in an answer to the second part
of the problem, yes the two triangles are similar.