### Video Transcript

The figure shows two triangles
π΄π΅πΆ and π΄ prime π΅ prime πΆ prime. Firstly, describe the single
transformation that would map π΄π΅πΆ to π΄ prime π΅ prime πΆ prime. Secondly, hence, determine whether
triangles π΄π΅πΆ and π΄ prime π΅ prime πΆ prime are similar.

Firstly, weβre asked for the single
transformation that maps π΄π΅πΆ to π΄ prime π΅ prime πΆ prime. This word single is really
important. We canβt give a combination of
transformations as our answer because the question tells us that itβs possible to
achieve this with just one single transformation.

Letβs look at the triangles more
closely. The two triangles have different
sizes, which tells us that the type of transformation weβre 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
triangles.

The base of triangle π΄π΅πΆ is one
unit, whereas the base of triangle π΄ prime π΅ prime πΆ prime is two units. The perpendicular height of π΄π΅πΆ
is two, and the perpendicular height of π΄ prime π΅ prime πΆ prime is four. Therefore, the sides in triangle π΄
prime π΅ prime πΆ prime are always twice as long as the corresponding sides in
triangle π΄π΅πΆ. So the scale factor of the dilation
is two.

Finally, letβs 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 π΄ and π΄ prime. Next I draw in the line connecting
πΆ and πΆ prime. Now actually, it would be enough
just to draw these two rays, but Iβll draw the third in as well.

What youβll 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 π΄π΅πΆ to π΄ prime π΅
prime πΆ prime is the dilation from the point negative three, zero, with scale
factor two.

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.