### Video Transcript

Given that π΄π΅ is equal to πΆπ·,
ππΆ is equal to 10 centimeters, and π·πΉ is equal to eight centimeters, find the
length of line segment ππΈ.

In this question, we are trying to
calculate the length of ππΈ, which is the distance from the chord π΄π΅ to the
center of the circle π. We begin by recalling that two
chords of equal lengths are equidistant from the center. And in this question, we are told
that the two chords π΄π΅ and πΆπ· are equal in length. This means that the length ππΉ
must be equal to ππΈ. The line segment ππΈ
perpendicularly bisects the chord π΄π΅. Likewise, ππΉ is the perpendicular
bisector of πΆπ·. Since we are told π·πΉ is equal to
eight centimeters, πΆπΉ, π΄πΈ, and π΅πΈ are all also equal to eight centimeters.

Our next step is to consider the
right triangle ππΉπΆ. Using the Pythagorean theorem, ππΉ
squared plus πΆπΉ squared is equal to ππΆ squared. By subtracting πΆπΉ squared from
both sides and substituting in the values of πΆπΉ and ππΆ, we have ππΉ squared is
equal to 10 squared minus eight squared. This is equal to 36. Square rooting both sides of this
equation and knowing that ππΉ must be a positive answer, we have ππΉ is equal to
six. Since ππΉ is equal to six
centimeters, ππΈ must also be equal to six centimeters. The perpendicular distance from the
center of the chord π΄π΅ to the center of the circle π, which is the line segment
ππΈ, is equal to six centimeters.