### Video Transcript

In the figure below, line segment
π΅πΆ is a diameter of the circle π, π΄π΅ equals π΄πΆ, the measure of angle π΅π΄πΆ
equals 60 degrees, π΅π equals 22.9 centimeters, and line segment ππ is
perpendicular to line segment π΄π΅. Find the length of line segment
π΄πΈ.

We want to find how long line
segment π΄πΈ is. But first, letβs take the
additional information our question gave us and add it into our diagram. We know that π΅πΆ is a diameter of
this circle; itβs a chord that passes through the center. This means that line segment ππΆ
will be equal in length to line segment ππ΅. We also know that line segment π΄π΅
is equal in length to line segment π΄πΆ. If line segment π΄π΅ is equal to
line segment π΄πΆ, this means the angles opposite them will be equal in length. This is a property of
triangles.

Since we know that one of the
angles is 60 degrees and the other two are equal to each other, this will be an
equilateral triangle. After that, we see that π΅π equals
22.9 centimeters. And if we add an additional radius
from π to πΈ, we see that that will be equal to the radius from π to π΅. And then, we can say that line
segment ππ is a perpendicular bisector of πΈπ΅. And we can say that line segment
πΈπ must then also be equal in measure to 22.9 centimeters.

But now, if we consider the fact
that this triangle, πΈππ΅, is an isosceles triangle, we can say that the angles
opposite the equal sides must be equal to each other, which means those angles must
both be 60 degrees. And, of course, if two angles in a
triangle are 60 degrees, then the third angle in the triangle must be 60
degrees. How does this help us? Well, it tells us that if we add
22.9 plus 22.9, weβre going to get the radius of the circle. And if from π to π΅ is 45.8
centimeters, then from πΆ to π is 45.8 centimeters. And this means to find the distance
from π΄ to πΆ, we just need to add 45.8 plus 45.8. That gives us 91.6 centimeters.

And since π΄π΅ is equal in length
to π΄πΆ, π΄π΅ must also measure 91.6 centimeters. But π΄π΅ is made up of two
segments, one π΄πΈ and one πΈπ΅. Remember π΄πΈ is our unknown value,
but πΈπ΅ will be 22.9 plus 22.9. πΈπ΅ is equal to 45.8
centimeters. And if 91.6 centimeters is equal to
π΄πΈ plus 45.8 centimeters, then we subtract 45.8 from both sides. We get that line segment π΄πΈ must
be equal to 45.8 centimeters. What this tells us about our circle
is that point πΈ divided line segment π΄π΅ in half.