Video Transcript
Given that π΄π΅πΆπ· is similar to
ππππΏ, find the measure of angle ππΏπ and the length of the line segment
πΆπ·.
We are told here that π΄π΅πΆπ· is
similar to ππππΏ. And that means that their
corresponding angles are congruent and their corresponding sides are in
proportion. We are first asked to find the
measure of angle ππΏπ. But we noticed that we donβt have
enough information about the angles in polygon ππππΏ to work out this angle
measure. However, as these polygons are
similar, we know that this angle measure of π΅πΆπ· is corresponding to angle πππΏ,
and so the angle πππΏ must have a measure of 85 degrees. We can then use the fact that the
internal angle measures in a quadrilateral add to 360 degrees to work out that the
measure of angle ππΏπ must be equal to 360 degrees subtract 85 degrees plus 109
degrees plus 105 degrees. This gives us 61 degrees.
Next, we need to find the length of
the line segment πΆπ·. We note that the length of πΆπ·
corresponds with the length ππΏ in polygon ππππΏ. The proportion of these sides will
be the same proportion as that between all other pairs of corresponding sides in the
polygon. When we are answering problems
involving similar polygons, we need to look for or be able to calculate another pair
of corresponding sides. Here we are given that π΄π΅ is 75
centimeters and ππ is 150 centimeters. We could therefore write that πΆπ·
over ππΏ is equal to π΄π΅ over ππ.
Filling in the length measurements,
we have that πΆπ· over 246.2 is equal to 75 over 150. We can take out our common factor
of 75 from the numerator and denominator on the right-hand side. And then we can rearrange by
multiplying through by 246.2. The length of πΆπ· is therefore
123.1 centimeters. Alternatively, we couldβve found
the scale factor or ratio of enlargement between the two polygons. Using the side length of π₯π¦ and
π΄π΅, we couldβve found that to go to the polygon π΄π΅πΆπ· from the polygon
ππππΏ, we multiply by one-half. Multiplying the length 246.2
centimeters by one-half would have given us the same value of 123.1 centimeters.
As we have now answered both parts
of this question, we can give the answer that the measure of angle ππΏπ is 61
degrees and the length of πΆπ· is 123.1 centimeters.