### Video Transcript

In this video, we will learn how to
use the properties of similar polygons to find unknown angles, side lengths, scale
factors, and perimeters. Firstly, we recall that a polygon
is a closed shape with straight sides. So, for example, a triangle, a
square, and a hexagon are all examples of polygons, whereas a circle isnโt as its
sides are not all straight. Two polygons are said to be
mathematically similar if one is an enlargement of the other. This means that two key things will
be true.

Firstly, corresponding angles in
the two similar polygons are congruent or identical. Secondly, corresponding pairs of
sides are in proportion. Another way of saying this is to
say that corresponding pairs of sides are all in the same ratio. So in our example here, the length
of ๐ด๐ต divided by the length of the corresponding side ๐ธ๐น gives the same result
as if we divide ๐ต๐ถ by ๐น๐บ, and if we divide ๐ถ๐ท by ๐บ๐ป, and if we divide ๐ท๐ด
by ๐ป๐ธ.

If two polygons are similar, then
we can express this using the notation on screen. Notice that the order of the
letters is important. So if Iโve said ๐ด๐ต๐ถ๐ท is similar
to ๐ธ๐น๐บ๐ป, then this means that the angle at ๐ด corresponds to the angle at
๐ธ. And the side ๐ถ๐ท corresponds to
the side ๐บ๐ป. If two polygons are similar, then
we can calculate the scale factor between them. This is the ratio of corresponding
lengths and will always be a multiplier. Itโs the value we need to multiply
the length in one polygon by to find the corresponding length in the other
polygon. We can calculate the scale factor
in either direction.

For example, if we need to multiply
the lengths of ๐ด๐ต๐ถ๐ท by ๐ in order to give the corresponding lengths in
๐ธ๐น๐บ๐ป, then in order to go the other way, weโd need to multiply the lengths of
๐ธ๐น๐บ๐ป by one over ๐ to give the corresponding lengths in ๐ด๐ต๐ถ๐ท. Itโs important to notice that the
scale factor is always a multiplier. So, in formulae, we may think of
going from the larger polygon to the smaller polygon as dividing by ๐. But our scale factor will be the
multiplier one over ๐. Letโs now look at a question where
we need to calculate the scale factor between two similar polygons and then
determine the length of a side.

Given that the rectangles shown are
similar, what is ๐ฅ?

Weโre told in the question that
these two rectangles are mathematically similar, which means that two things are
true. Firstly, all pairs of corresponding
angles between the two rectangles are congruent. Now, for a pair of rectangles, this
is true even for nonsimilar rectangles as all the interior angles in a rectangle are
90 degrees. Secondly, and more importantly
here, all pairs of corresponding sides are in proportion.

Weโre looking to find the value of
๐ฅ which represents a side length in the larger rectangle. We therefore need to know the scale
factor ๐ that is the multiplier that takes us from the smaller rectangle to the
larger. We can use any pair of
corresponding sides to work out the scale factor. Itโs equal to the new length
divided by the original length. From the figure, we can see that we
have a pair of corresponding sides of 29 centimeters on the smaller rectangle and 58
centimeters on the larger. We therefore divide the new length
of 58 by the original length of 29, giving a scale factor of 58 over 29, which
simplifies to two.

Remember, the scale factor is
always a multiplier. So this tells us that the lengths
on the larger rectangle are all twice the corresponding lengths on the smaller
rectangle. To work out the value of ๐ฅ then,
we need to take the corresponding length on the smaller rectangle, 26, and multiply
it by the scale factor of two, which gives 52. So we found the value of ๐ฅ. If we had wanted to calculate a
length on the smaller rectangle rather than one on the larger rectangle, we couldโve
used the scale factor of one-half. Thatโs the reciprocal of two. This couldโve been found by
dividing the length of 29 centimeters by the corresponding length of 58
centimeters.

We can check our value for ๐ฅ by
multiplying it by one-half, so 52 multiplied by one-half or half of 52, which is
indeed equal to 26. In formulae, we may think of this
as dividing by two. But remember, the scale factor is
always a multiplier, so weโre using a multiplier of one-half.

Letโs consider another example.

If the two following polygons are
similar, find the value of ๐ฅ.

Weโre told that these two polygons
are mathematically similar, which means that two key things are true. Firstly, corresponding pairs of
angles are congruent. And secondly, corresponding pairs
of sides are in proportion or in the same ratio. We want to calculate this ratio or
scale factor. And in this case, as ๐ฅ represents
the length on the smaller polygon, weโre working in this direction. The scale factor can be calculated
from any corresponding pair of sides by dividing the new length by the original
length.

We have a corresponding pair of
side lengths of 34 centimeters and 85 centimeters. So dividing the new length, thatโs
the length which is in the same polygon as the one we want to calculate, by the
original length, we have a scale factor of 34 over 85. This can be simplified by dividing
both the numerator and denominator by 17 to give two-fifths. Notice that this value feels
right. Weโre moving from the larger
polygon to the smaller one. So the length should be getting
smaller.

Our scale factor is a fractional
value less than one. And when we multiply something by a
fractional value less than one, such as two-fifths, it will become smaller. To calculate the value of ๐ฅ then,
we need to take the corresponding length on the larger polygon, which is 75
centimeters, and multiply it by our scale factor of two-fifths. We can simplify this calculation by
canceling a factor of five from both the numerator and denominator to give ๐ฅ equals
15 multiplied by two, which is of course 30. So weโve found the value of ๐ฅ.

We can check our answer by
confirming that the ratio of corresponding pair of sides is indeed the same. We have 30 over 75 and 34 over
85. Both fractions do indeed cancel to
our scale factor of two-fifths. And so this confirms that our
answer is correct.

In our next example, weโll see how
we can test whether or not two polygons are similar.

Is polygon ๐ด๐ต๐ถ๐ท similar to
polygon ๐บ๐น๐ธ๐?

From the figure, we can see that
the two polygons weโve been given are each parallelograms. So we can conclude that they are at
least the same type of shape to begin with. To determine whether theyโre
similar, we need to test two things. Firstly, we need to test whether
corresponding pairs of angles are congruent. And secondly, we need to test
whether corresponding pairs of sides are in proportion or in the same ratio.

Now, itโs important to remember
that when weโre working with similar polygons, the order of the letters is
important. So if these polygons are similar,
then the angle at ๐ด will correspond to the angle at ๐บ. The angle at ๐ต will correspond to
the angle at ๐น, and so on. We can therefore deduce that the
polygons have been drawn in the same orientation.

Letโs consider the angles first of
all then. In polygon ๐บ๐น๐ธ๐, weโve been
given a marked angle of 110 degrees. And in polygon ๐ด๐ต๐ถ๐ท, weโve been
given a marked angle of 70 degrees. One thing we do know about
parallelograms is that their opposite angles are equal. So in parallelogram ๐บ๐น๐ธ๐, the
angle at ๐ will be 110 degrees. And in parallelogram ๐ด๐ต๐ถ๐ท, the
angle at ๐ถ will be 70 degrees.

Letโs consider the angle at ๐บ in
the polygon ๐บ๐น๐ธ๐. By extending the line ๐น๐บ, we now
see that we have two parallel lines ๐๐บ and ๐ธ๐น and a transversal ๐บ๐น. We know that corresponding angles
in parallel lines are equal, which means that the angle above the line ๐๐บ will be
equal to the angle above the line ๐ธ๐น. Itโs 110 degrees. We also know that angles on a
straight line sum to 180 degrees, which means the angle below ๐๐บ will be 180 minus
110. Itโs 70 degrees. This shows us that the angle at ๐บ
in polygon ๐บ๐น๐ธ๐ is equal to the angle at ๐ด in polygon ๐ด๐ต๐ถ๐ท.

We already said that opposite
angles in parallelograms are equal. So the angle at ๐ธ is also 70
degrees, which is equal to the angle at ๐ถ in polygon ๐ด๐ต๐ถ๐ท. We could use the same logic in
parallelogram ๐ด๐ต๐ถ๐ท to show that the angles at ๐ต and ๐ท are each 110 degrees,
which are the same as the angles at ๐น and ๐ in the larger polygon. Weโve shown then that all pairs of
corresponding angles are indeed congruent. So our answer to the first check is
yes.

Letโs now consider whether
corresponding pairs of sides are in proportion. Firstly, from the figure, we can
see weโve been given side length ๐ด๐ต; itโs 13 centimeters. And if the two polygons are
similar, this will correspond to ๐บ๐น. We havenโt been given the length of
๐บ๐น. But we know that opposite sides in
a parallelogram are equal in length. So it will be the same as ๐๐ธ. Comparing the ratio of these two
sides then, we find that ๐ด๐ต over ๐บ๐น is equal to 13 over 26, which simplifies to
one-half.

The other pair of potentially
corresponding sides weโve been given are ๐ต๐ถ and ๐ธ๐น. The ratio here is 11.5 over 23,
which again simplifies to one-half. Now, Iโve written ๐ธ๐น here. But really, if weโre to be
consistent with the order of letters, then we should really write ๐น๐ธ as point ๐น
corresponds to point ๐ต and point ๐ธ corresponds to point ๐ถ. However, in calculating the ratio,
the length ๐ธ๐น is of course the same as the length ๐น๐ธ. So it makes no practical
difference.

The ratio of these pairs of
corresponding sides is therefore the same. As opposite sides in a
parallelogram are equal in length, the same will be true for the remaining two pairs
of corresponding sides. And so the answer to our second
check, โare corresponding pairs of sides in proportion?โ, is also yes. Hence, both of the criteria for
these two polygons to be similar are fulfilled. And so we can answer, yes, polygon
๐ด๐ต๐ถ๐ท is similar to polygon ๐บ๐น๐ธ๐.

Letโs now consider another example
in which we prove similarity of two polygons.

Are these two polygons similar? If yes, find the scale factor from
๐๐๐๐ฟ to ๐ด๐ต๐ถ๐ท.

To determine whether these two
polygons are similar, we need to consider two questions. Firstly, are corresponding pairs of
angles congruent? And secondly, are corresponding
pairs of sides in proportion? Remember that the order of the
letters is important. So if these two polygons are
similar, then the angle at ๐ will correspond with the angle at ๐ด. And, for example, the side ๐๐ฟ
will correspond with the side ๐ถ๐ท.

Letโs consider the angles
first. And looking at the figure, we can
see that three of the angles in each polygon have been marked using different
notation. The single line marking the angle
at ๐ฟ and the angle at ๐ท indicates that these two angles are equal. The pair of lines at angle ๐ and
angle ๐ด indicates that these two angles are equal. In the same way, the triple line at
angle ๐ and angle ๐ถ indicates that these two angles are equal. But what about the final angle in
each polygon? Well, each of these polygons have
four sides. They are quadrilaterals. And we know that the sum of the
interior angles in any quadrilateral is 360 degrees.

If all of the other three angles in
the two quadrilaterals are the same, then in each case, we will have the same amount
left over from the total of 360 degrees to form the final angle. And therefore, we can also say that
angle ๐ is equal to angle ๐ต. So weโve seen that all pairs of
corresponding angles are indeed congruent between the two polygons. So the answer to our first check is
yes.

Next, letโs consider the sides. And we need to check whether
corresponding pairs of sides are in proportion. That is, do we get the same value
when we divide each side of one polygon by the corresponding side of the other? Weโve been given all of the side
lengths we need. So we can substitute them and work
out what each of these ratios simplify to. In fact, they are all equal to
four-fifths or 0.8. And so corresponding pairs of sides
are indeed in proportion. As weโve answered yes to both
statements, we can conclude that the two polygons are indeed similar.

Weโre now asked to find the scale
factor from ๐๐๐๐ฟ to ๐ด๐ต๐ถ๐ท. Thatโs traveling in this
direction. Remember, the scale factor is the
value we multiply the lengths on the first polygon by in order to give the
corresponding lengths on the second. And in fact, weโve already worked
this out. Itโs the value we get when we
divide a new length, thatโs a length on the polygon weโre going to, by an original
length. Thatโs the corresponding length on
the polygon weโre coming from.

Weโve already seen that when we
divide each length on ๐ด๐ต๐ถ๐ท by the corresponding length on ๐๐๐๐ฟ, we get the
value four-fifths. So this is our scale factor. Notice that this value does make
logical sense. The lengths on ๐ด๐ต๐ถ๐ท are smaller
than the lengths on ๐๐๐๐ฟ. So our scale factor should be a
fractional value less than one. So weโve completed the problem. We found that these two polygons
are similar. And the scale factor going from
๐๐๐๐ฟ to ๐ด๐ต๐ถ๐ท, thatโs from the larger polygon to the smaller, as a decimal is
0.8.

Just as a reminder, if we were
going back the other way, so finding the scale factor from ๐ด๐ต๐ถ๐ท to ๐๐๐๐ฟ,
then we would have the reciprocal. The reciprocal of a fraction can be
found by flipping it, so swapping its numerator and denominator. The scale factor going in the other
direction would be the multiplier of five over four. Again, this makes sense. Five over four is a little bit
bigger than one. Itโs 1.25 as a decimal. And so weโre multiplying the
lengths on ๐ด๐ต๐ถ๐ท by something bigger than one, which will give larger values for
the corresponding lengths on ๐๐๐๐ฟ.

Letโs now summarize some of the key
points from this video. Firstly, two polygons are similar
if corresponding pairs of angles are congruent and all corresponding pairs of sides
are in proportion. To calculate the scale factor
between two similar polygons, we can use the lengths of any pair of corresponding
sides. And we divide the new length,
thatโs the length on the polygon weโre going to, by the corresponding length on the
original. Thatโs the polygon weโre coming
from.

Remember that the scale factor is
always a multiplier. When weโre going from the smaller
shape to the larger shape, we will always have a scale factor greater than one. When going in the other direction,
so going from the larger shape to the smaller shape, the scale factor will be the
reciprocal of the scale factor working in the other direction. It will be one over ๐. And in this case, the scale factor
will be less than one. In formulae, we may think of this
as dividing by the previous scale factor of ๐. But our scale factor should really
be a multiplier.

Once we know that two polygons are
similar, we can calculate the measures of any missing angles. And we can use scale factors to
calculate the lengths of any missing sides, provided weโve been given the
corresponding information on the other polygon.