In the given figure, what
combination of transformations would map circle 𝐴 onto circle 𝐵? Is it (A) a translation of four
left and six down, followed by a dilation of scale factor one-half? Is it (B) a translation of six left
and four down, followed by a dilation of scale factor two-thirds? (C) A translation of four left and
six down, followed by a dilation of scale factor two-thirds. (D) A translation of six right and
four up, followed by a dilation of scale factor three over two. Or is it (E) a translation of four
left and six up, followed by a dilation of scale factor three-quarters?
So, in this question, we’re looking
to describe a combination of transformations. And that’s great because we can
include more than one transformation when we’re describing this. We’re looking to describe the
transformations that map circle 𝐴 onto circle 𝐵. Well, 𝐴 is the larger circle, and
𝐵 is the smaller. And so, that’s a little bit of a
hint as to what transformation we might consider.
The four transformations, sometimes
called similarity transformations, that we’re going to look at are rotations,
reflections, dilations, and translations. When we rotate a shape, we turn
it. The use of the letter “t ” here is
a reminder of that. When we reflect it, the “fl ”
reminds us we flip the shape in a mirror line. Dilation, which is sometimes called
enlargement, is where we make a shape larger, that’s the “l, ” or sometimes
smaller. And when we translate a shape, the
“sl ” reminds us that we slide it. And when we do so, we slide it left
or right and up or down.
We said we were mapping the larger
circle onto the smaller circle. So, there’s an indication to us
that we’re definitely going to be dilating the shape. But also we notice that the center
of the circle 𝐴, which is currently at the point with coordinates six, zero, maps
onto the center of circle 𝐵 with coordinates zero, negative four. So, how do we achieve that? Well, we’re going to need to slide
the entire circle, and so we’re going to be translating it as well.
Let’s deal with the translation
first. We said the translation slides the
center of circle 𝐴 onto the center of circle 𝐵. Note that we’ve chosen the center
of the circle rather than a point on its circumference because it’s the only
constant when we dilate it. And so, how many units left or
right are we going to need to slide the center? Well, the 𝑥-coordinate moves from
six to zero. So, we’re going to be moving this
six units to the left. Then, the 𝑦-coordinate goes from
zero to negative four, so we slide it four down. And we, therefore, have a
translation of six left and four down.
So, that’s the translation part,
but what about the dilation? Well, the shape has gotten
smaller. So we’re going to have a fractional
scale factor between zero and one. And we might recall that we can
calculate a scale factor by dividing a dimension on the new shape by a corresponding
dimension on the original shape. So, let’s consider the radius of
our circle. I’ve drawn that in on circle
𝐴. And here it is on circle 𝐵. Now, remember, the radius is simply
the length of the line that joins the center of the circle to its circumference. So it doesn’t matter which radius
On shape 𝐵, the radius is two
units, whereas on shape 𝐴, it’s three units. Since we’re mapping 𝐴 onto 𝐵, we
take the radius of circle 𝐵 and we divide that by the radius of circle 𝐴. So, the scale factor must be
two-thirds. And so, we’re ready to fully
describe the combination of transformations.
We have a translation of six units
left and four down, followed by a dilation or an enlargement of scale factor
two-thirds. Comparing that to the answers given
here, we see the correct answer is (B), a translation of six left and four down,
followed by a dilation of scale factor two-thirds.