Video Transcript
In the given figure, what
combination of transformations would map circle 𝐴 onto circle 𝐵? Is it (A) a translation of two
right and six down followed by a dilation of scale factor two? (B) A translation of two left and
six up followed by a dilation of scale factor two. (C) A translation of four left and
two down followed by a dilation of scale factor one. (D) A translation of six left and
two up followed by a dilation of scale factor one. Or is it (E) a translation of six
right and two down followed by a dilation of scale factor two?
So, we’re looking for a combination
of transformations. So, that tells us we’re going to be
describing more than one. And we’re mapping circle 𝐴, that’s
the circle with center at negative three, three, onto circle 𝐵, that’s the larger
circle with center at three, one.
Let’s remind ourselves the four
transformations we tend to use. They are reflections, rotations,
dilations, and translations. The “fl” in the word reflection
remind us that when we reflect a shape, we flip it over a mirror line. When we rotate a shape, we “t” turn
it about a center in a given direction and angle. When we dilate a shape, which is
sometimes called enlarging it, we make it either larger or smaller. And finally, when we translate a
shape, we slide it.
So, let’s look at what we’ve done
to map circle 𝐴 onto circle 𝐵. And we’re going to use the center
as the point of reference. Firstly, we see that to map circle
𝐴 onto circle 𝐵, we need to move the entire circle right a number of units. In fact, we need to move it right
six units. We then move it down two units. And so, we can see that we’re
actually beginning with a slide. We’re beginning with a translation
six units right and two units down.
But then what happens? We said earlier that circle 𝐵 is
much larger than circle 𝐴. And so we must be dilating or
enlarging the circle. When we enlarge or dilate a shape,
we describe it in terms of its scale factor. And the scale factor is calculated
by dividing a length on the new shape by the corresponding length on the old
shape. So, in this case, let’s look at the
radii of our circles.
The radius is a line joining the
center to a point on the circumference. So we can actually choose any point
here. On our larger circle, the radius is
two units in length. And on our smaller original circle,
the radius is one unit. So, the scale factor for
enlargement or dilation here must be two divided by one, which is simply two.
And so, we see that we have a
translation six units right and two units down followed by a dilation with a scale
factor of two. Let’s compare this to the options
given in our question. When we do, we see that the correct
answer is (E). It’s a translation of six right and
two down followed by a dilation of scale factor two.
