### Video Transcript

If π΄π΅πΆπ· is similar to πΈπΉπΊπ»,
find the scale factor from π΄π΅πΆπ· to πΈπΉπΊπ» and the values of π and π.

In this problem, we are told that
the two given quadrilaterals, π΄π΅πΆπ· and πΈπΉπΊπ», are similar. We can recall that two polygons are
similar if their corresponding angles are congruent and their corresponding sides
are in proportion. We can recognize that similar
polygons can be considered as a dilation of each other. If the scale factor is one, the
polygons could instead be called congruent.

Here, the first part of the
question asks us to find the scale factor of similarity. This is equivalent to finding the
proportion that corresponding sides are in. So, letβs see if we can identify a
corresponding pair of sides in the diagrams for which we know both their
lengths.

We have side π΅πΆ given as 47
centimeters and side πΉπΊ given as 18.8 centimeters. Now, because we are told to find
the scale factor from π΄π΅πΆπ· to πΈπΉπΊπ», then the way in which we write the
proportion as a fraction is very important. Because we are going to πΈπΉπΊπ»,
then the side πΉπΊ from this quadrilateral is on the numerator. And because we are coming from
π΄π΅πΆπ·, then side π΅πΆ is written on the denominator. Once we have that, then we can fill
in the respective lengths of πΉπΊ and π΅πΆ as 18.8 and 47 centimeters.

Next, we need to simplify this
fraction. If we are doing this without the
use of a calculator, then often it is easiest to get rid of the decimal value of
18.8. So, multiplying both the numerator
and denominator by 10 would give us 188 over 470. We might then choose to halve each
of the numerator and denominator to give 94 over 235 and then divide each of these
by 47. Or alternatively, we may have
divided 188 and 470 by 94 and arrived at this same fully simplified value of
two-fifths.

By using either method, we have
found that the proportion of the corresponding sides is two-fifths. And it is also the scale factor
from π΄π΅πΆπ· to πΈπΉπΊπ». That means if we took any of the
side lengths in π΄π΅πΆπ· and multiplied it by two-fifths, we would get the
corresponding side length in πΈπΉπΊπ». Knowing this value will now allow
us to determine the values of π and π.

Letβs identify that π is the
length of the side πΊπ». And the corresponding side in
π΄π΅πΆπ· is πΆπ·. And so, if we travel from π΄π΅πΆπ·
to πΈπΉπΊπ», weβll be multiplying 34 centimeters by two-fifths to give π
centimeters. As a fraction, we can calculate the
left-hand side of 34 times two-fifths as 68 over five. And writing this as a mixed number
fraction, we have 13 and three-fifths equals π. As a decimal then, we have 13.6
equals π. So, we have determined that the
value of π is 13.6.

Next, letβs calculate the value of
π. π can be found as part of the
length of π΄π΅, and the corresponding side on πΈπΉπΊπ» is the side πΈπΉ. Now, we could determine the length
of π΄π΅ using the scale factor of two-fifths, or we could find the scale factor in
the opposite direction. That is, what is the scale factor
from πΈπΉπΊπ» to π΄π΅πΆπ·? Well, if we have a scale factor in
one direction and we want to find the scale factor in the opposite direction, we
calculate its reciprocal. The reciprocal of two-fifths is
five over two. So, the scale factor from πΈπΉπΊπ»
to π΄π΅πΆπ· is five over two.

So, we can write that the side
πΈπΉ, which is 19.2, multiplied by five over two equals side π΄π΅, which is π plus
four. We can then simplify the left-hand
side as 96 over two. Then, we have the equation 48 is
equal to π plus four. Finally, subtracting four from both
sides, we can calculate that π is equal to 44, which is the final part of the
question answered.

We can give the answers to all
three parts of the question as the scale factor equals two-fifths, π equals 13.6,
and π equals 44.