### Video Transcript

Given that ππΆ is equal to ππΉ,
which is equal to three centimeters, π΄πΆ is equal to four centimeters, line segment
ππΆ is perpendicular to line segment π΄π΅, and line segment ππΉ is perpendicular
to line segment π·πΈ, find the length of the line segment π·πΈ.

We are told in the question that
ππΆ is equal to ππΉ. And this means that the two chords
π΄π΅ and π·πΈ are equidistant from the center π. They are both three centimeters
away from the center. We recall the theorem that states
that if two chords are equidistant from the center, they are also equal in
length. This means that in this question,
the length π΄π΅ is equal to the length π·πΈ. We also know that ππΆ is the
perpendicular bisector of π΄π΅. And this means that π΄π΅ is equal
to two multiplied by π΄πΆ. Since π΄πΆ is equal to four
centimeters, π΄π΅ is equal to eight centimeters. We can therefore conclude that the
length of the chord π·πΈ is eight centimeters.

So far in this video, we have
discussed the lengths of chords depending on their distance from the center of the
circle. We will now consider the converse
relationship. In the diagram shown, we have two
congruent circles. We will consider the case where the
chords π΄π΅ and πΆπ· are equal in length and, more importantly, what this says about
the distance of the chords from their respective centers. In this example, these are the
lengths ππΈ and ππΉ, respectively, Adding the radii ππ΄ and ππΆ, we know that
these lengths must be equal, as the circles are congruent. As the chords are equal in length,
the half chords must also be equal in length such that π΄πΈ is equal to πΆπΉ. This is because π΄πΈ is equal to a
half of π΄π΅ and πΆπΉ is equal to a half of πΆπ·.

Using our knowledge of the
Pythagorean theorem, the third sides of our right triangles must also be equal in
length. The length ππΈ is equal to the
length ππΉ. In other words, the distance of the
chords from their respective centers are equal. This can be summarized as
follows. Two chords of equal lengths in the
same circle or congruent circles are equidistant from the center of the circle or
the respective centers.