### Video Transcript

If the line π΅πΉ is perpendicular
to the lines π΄π΅, πΆπ·, and πΈπΉ, find the length of π΄πΆ, where π΅π· equals π·πΉ
equals three centimeters and π΄πΈ equals 12 centimeters.

Let us start off by analyzing the
information we have been given. First of all, weβve been told that
the line π΅πΉ is perpendicular to each of the lines π΄π΅, πΆπ·, and πΈπΉ, which is
indicated by the right angles on the diagram. We have also been given that line
segments π΅π· and π·πΉ have the same length of three centimeters. And finally, we know that line
segment π΄πΈ is 12 centimeters.

The length that we need to find is
that of π΄πΆ, which is part of this line segment. Now, since all three of the
horizontal lines are perpendicular to the same line, this means that they must be
parallel to each other. That is, lines π΄π΅, πΆπ·, and πΈπΉ
are parallel.

From here, we can recall the
property that if a set of parallel lines divide a transversal into segments of equal
length, then they divide any other transversal into segments of equal length. Therefore, since π΅π· is a
transversal of the three parallel lines and π΅π· and π·πΉ are segments of equal
length, we can apply this property.

Thus, the transversal π΄πΈ will
also be divided into segments of equal length by these lines. So, π΄πΆ equals πΆπΈ. Now, we can use the fact that π΄πΈ
is 12 centimeters. So, we can conclude that π΄πΆ is
exactly half of π΄πΈ, which is half of 12, which is six.

Hence, adding in the correct units,
we can conclude that the length of π΄πΆ is six centimeters.