Video Transcript
If 𝐴𝐵𝐶𝐷𝐸 is similar to
𝑃𝑄𝑅𝑆𝑇, find the scale factor of 𝐴𝐵𝐶𝐷𝐸 to 𝑃𝑄𝑅𝑆𝑇 and the perimeter of
𝑃𝑄𝑅𝑆𝑇.
Now I said this automatically when
I read the question, but firstly just a reminder that this notation means “is
similar to.” So we’re being told that these two
polygons, which are pentagons, are similar to one another. This means that two things are
true. Corresponding pairs of angles
between the two polygons are congruent, and corresponding pairs of sides are in the
same ratio. Now we do need to be a little
careful here when looking at the two figures because we know that the order of
letters is important when discussing similar polygons.
So if 𝐴𝐵𝐶𝐷𝐸 is similar to
𝑃𝑄𝑅𝑆𝑇, then this means that the side 𝐴𝐵, for example, corresponds to the side
𝑃𝑄. On the figure, we can see that
these two sides are actually in different positions on the two polygons. In fact, they’ve been presented as
reflections. So we just need to be careful when
matching up pairs of corresponding sides and pairs of corresponding angles. It actually won’t really make any
practical difference because each of these polygons are symmetrical. But it is important in general to
make sure we’re careful when we’re matching up pairs of corresponding sides and not
just to assume that the two polygons have been drawn in the same orientation.
Now we were asked to find the scale
factor of 𝐴𝐵𝐶𝐷𝐸 to 𝑃𝑄𝑅𝑆𝑇. That’s moving from the larger
polygon to the smaller polygon in this direction here. To calculate a scale factor, we
need to divide a new length by an original length. So we’re looking to divide a length
on 𝑃𝑄𝑅𝑆𝑇 by the corresponding length on 𝐴𝐵𝐶𝐷𝐸. Now on the figure, we can see that
𝑃𝑇 corresponds to 𝐴𝐸. And whilst we haven’t been given
the length of 𝐴𝐸 explicitly, we can see that it is the same as the length
𝐶𝐷. So it’s also 21 units. The scale factor then, dividing a
new length on 𝑃𝑄𝑅𝑆𝑇 by an original length on 𝐴𝐵𝐶𝐷𝐸, is 14 over 21. And dividing both the numerator and
denominator by seven, this simplifies to two-thirds.
This value makes sense
intuitively. We’re moving from the larger
polygon to the smaller one. And so our scale factor should be a
fractional value less than one, and two-thirds is. So we’ve answered the first part of
the question. And now we need to answer the
second, which is to calculate the perimeter of 𝑃𝑄𝑅𝑆𝑇. We therefore need to know all of
the side lengths.
We can work out the side length of
𝑅𝑆 straight away because it’s the same as the side length 𝑃𝑇; it’s 14 units. To calculate the other side
lengths, we need to use the scale factor we’ve just worked out. Side 𝑃𝑄 is corresponding with
side 𝐴𝐵. So we can calculate its length by
taking the length of 𝐴𝐵, 24, and multiplying by our scale factor of
two-thirds. We can cancel a factor of three
from the numerator and denominator, giving eight multiplied by two, which is of
course equal to 16.
So we have the length 𝑃𝑄. And on the diagram, we can see that
the length 𝑅𝑄 is the same as this. So it’s also 16 units. The final length we need is 𝑆𝑇,
which is corresponding with the side 𝐷𝐸 on the larger polygon. Notice that it is 𝐷𝐸 and not 𝐸𝐷
because the two polygons, remember, have been drawn as reflections of one
another. So to calculate 𝑆𝑇, we take the
length of 𝐷𝐸, which is 27, and multiply it by our scale factor of two-thirds. Once again, we can cancel a factor
of three, giving 𝑆𝑇 equals nine multiplied by two, which is 18 units.
We now have the lengths of all of
the sides of 𝑃𝑄𝑅𝑆𝑇. So we’re able to calculate its
perimeter. It’s equal to 16 plus 16 plus 14
plus 18 plus 14, which is 78 units. We can conclude then that for these
two similar polygons, the scale factor of 𝐴𝐵𝐶𝐷𝐸 to 𝑃𝑄𝑅𝑆𝑇, so from the
larger polygon to the smaller, is two-thirds. And the perimeter of 𝑃𝑄𝑅𝑆𝑇 is
78 units.
