### Video Transcript

The figure shows a circle with
center π. Given that πΈπ΅ equals 42, πΈπΆ
equals 21, and πΈπ΄ equals 42, find the length of segment πΈπ· and then the radius
of the circle.

From the information given, we can
deduce that chord π΅π΄ is bisected, because its two parts πΈπ΅ and πΈπ΄ both equal
42. It is therefore relevant to recall
the first part of the chord bisector theorem. This theorem tells us that if a
straight line passing through the center of a circle bisects a chord in the same
circle, the line must be perpendicular to the chord. Therefore, the segment πΆπ·, which
passes through the center of our circle, is actually the perpendicular bisector of
chord π΅π΄.

Because chords π΅π΄ and πΆπ· are
perpendicular, we know that they form four right angles, one of which is the angle
π΄πΈπ. To answer this question, we must
find the length of segment πΈπ· and the radius. We note that segment ππ· is a
radius of the circle.

The only other piece of information
we have not considered yet is the length of segment πΈπΆ. We are told that πΈπΆ equals
21. We see from the diagram that
segment ππΆ is also a radius of the circle. And the length of ππΆ equals the
sum of ππΈ plus πΈπΆ. Since we donβt know the length
ππΈ, we will let ππΈ equal π₯. Then, we can say that ππΆ equals
π₯ plus 21. Since we know that all radii in a
circle have equal length, the length of any radius of our circle can be represented
by the eπ₯pression π₯ plus 21. Therefore, radius ππ· equals π₯
plus 21.

We also recognize segment ππ΄ as a
radius. So ππ΄ equals π₯ plus 21. At this point, we have an
eπ₯pression for each side of a right triangle, specifically triangle π΄πΈπ, which
we have highlighted in green. The length of side ππΈ equals π₯,
πΈπ΄ equals 42, and ππ΄ equals π₯ plus 21.

From here, we can use the
Pythagorean theorem, which states that the sum of the squares of the legs of a right
triangle equal the square of the hypotenuse. Substituting the eπ₯pressions for
each side length, we have π₯ squared plus 42 squared equals π₯ plus 21 squared. We will proceed to solve this
equation for π₯. Once we know the value of π₯, we
will be able to find the length of segment πΈπ· and the radius. We find that 42 squared is 1764 and
π₯ plus 21 squared is π₯ squared plus 42π₯ plus 441.

Now it is necessary to simplify our
quadratic equation into standard forms that it can be solved. However, after subtracting π₯
squared from each side of the equation, we realize that this is no longer a
quadratic equation. What remains is the equation 1764
equals 42π₯ plus 441. Solving this equation, we find that
π₯ equals 31.5.

Now we are ready to find the exact
length of the radius, which we previously found to be represented by the expression
π₯ plus 21. So we substitute 31.5 for π₯. And we find that the radius equals
52.5. Finally, to find the length of
segment πΈπ·, we will use the fact that πΈπ· equals ππΈ plus ππ·, which we see is
true from the diagram. ππΈ equals π₯. And because ππ· is a radius, it
equals 52.5. After substituting 31.5 for π₯ and
adding 52.5, we find that πΈπ· equals 84.

In conclusion, the length of
segment πΈπ· is 84, and the radius equals 52.5.