Video Transcript
If 𝐴𝐵𝐶𝐷 is similar to 𝐸𝐹𝐺𝐻,
find the scale factor from 𝐴𝐵𝐶𝐷 to 𝐸𝐹𝐺𝐻 and the values of 𝑋 and 𝑌.
In this problem, we are told that
the two given quadrilaterals, 𝐴𝐵𝐶𝐷 and 𝐸𝐹𝐺𝐻, 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, let’s see if we can identify a
corresponding pair of sides in the diagrams for which we know both their
lengths.
We have side 𝐵𝐶 given as 47
centimeters and side 𝐹𝐺 given as 18.8 centimeters. Now, because we are told to find
the scale factor from 𝐴𝐵𝐶𝐷 to 𝐸𝐹𝐺𝐻, then the way in which we write the
proportion as a fraction is very important. Because we are going to 𝐸𝐹𝐺𝐻,
then the side 𝐹𝐺 from this quadrilateral is on the numerator. And because we are coming from
𝐴𝐵𝐶𝐷, then side 𝐵𝐶 is written on the denominator. Once we have that, then we can fill
in the respective lengths of 𝐹𝐺 and 𝐵𝐶 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 𝐴𝐵𝐶𝐷 to 𝐸𝐹𝐺𝐻. That means if we took any of the
side lengths in 𝐴𝐵𝐶𝐷 and multiplied it by two-fifths, we would get the
corresponding side length in 𝐸𝐹𝐺𝐻. Knowing this value will now allow
us to determine the values of 𝑋 and 𝑌.
Let’s identify that 𝑋 is the
length of the side 𝐺𝐻. And the corresponding side in
𝐴𝐵𝐶𝐷 is 𝐶𝐷. And so, if we travel from 𝐴𝐵𝐶𝐷
to 𝐸𝐹𝐺𝐻, we’ll be multiplying 34 centimeters by two-fifths to give 𝑋
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 𝑋. As a decimal then, we have 13.6
equals 𝑋. So, we have determined that the
value of 𝑋 is 13.6.
Next, let’s calculate the value of
𝑌. 𝑌 can be found as part of the
length of 𝐴𝐵, and the corresponding side on 𝐸𝐹𝐺𝐻 is the side 𝐸𝐹. Now, we could determine the length
of 𝐴𝐵 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 𝐸𝐹𝐺𝐻 to 𝐴𝐵𝐶𝐷? 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 𝐸𝐹𝐺𝐻
to 𝐴𝐵𝐶𝐷 is five over two.
So, we can write that the side
𝐸𝐹, which is 19.2, multiplied by five over two equals side 𝐴𝐵, which is 𝑌 plus
four. We can then simplify the left-hand
side as 96 over two. Then, we have the equation 48 is
equal to 𝑌 plus four. Finally, subtracting four from both
sides, we can calculate that 𝑌 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, 𝑋 equals 13.6,
and 𝑌 equals 44.
