Video Transcript
The figure shows a circle with
center 𝑀. Given that 𝐸𝐵 equals 42, 𝐸𝐶
equals 21, and 𝐸𝐴 equals 42, find the length of segment 𝐸𝐷 and then the radius
of the circle.
From the information given, we can
deduce that chord 𝐵𝐴 is bisected, because its two parts 𝐸𝐵 and 𝐸𝐴 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 𝐶𝐷, which
passes through the center of our circle, is actually the perpendicular bisector of
chord 𝐵𝐴.
Because chords 𝐵𝐴 and 𝐶𝐷 are
perpendicular, we know that they form four right angles, one of which is the angle
𝐴𝐸𝑀. To answer this question, we must
find the length of segment 𝐸𝐷 and the radius. We note that segment 𝑀𝐷 is a
radius of the circle.
The only other piece of information
we have not considered yet is the length of segment 𝐸𝐶. We are told that 𝐸𝐶 equals
21. We see from the diagram that
segment 𝑀𝐶 is also a radius of the circle. And the length of 𝑀𝐶 equals the
sum of 𝑀𝐸 plus 𝐸𝐶. Since we don’t know the length
𝑀𝐸, we will let 𝑀𝐸 equal 𝑥. Then, we can say that 𝑀𝐶 equals
𝑥 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 e𝑥pression 𝑥 plus 21. Therefore, radius 𝑀𝐷 equals 𝑥
plus 21.
We also recognize segment 𝑀𝐴 as a
radius. So 𝑀𝐴 equals 𝑥 plus 21. At this point, we have an
e𝑥pression for each side of a right triangle, specifically triangle 𝐴𝐸𝑀, which
we have highlighted in green. The length of side 𝑀𝐸 equals 𝑥,
𝐸𝐴 equals 42, and 𝑀𝐴 equals 𝑥 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 e𝑥pressions for
each side length, we have 𝑥 squared plus 42 squared equals 𝑥 plus 21 squared. We will proceed to solve this
equation for 𝑥. Once we know the value of 𝑥, we
will be able to find the length of segment 𝐸𝐷 and the radius. We find that 42 squared is 1764 and
𝑥 plus 21 squared is 𝑥 squared plus 42𝑥 plus 441.
Now it is necessary to simplify our
quadratic equation into standard forms that it can be solved. However, after subtracting 𝑥
squared from each side of the equation, we realize that this is no longer a
quadratic equation. What remains is the equation 1764
equals 42𝑥 plus 441. Solving this equation, we find that
𝑥 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
𝑥 plus 21. So we substitute 31.5 for 𝑥. And we find that the radius equals
52.5. Finally, to find the length of
segment 𝐸𝐷, we will use the fact that 𝐸𝐷 equals 𝑀𝐸 plus 𝑀𝐷, which we see is
true from the diagram. 𝑀𝐸 equals 𝑥. And because 𝑀𝐷 is a radius, it
equals 52.5. After substituting 31.5 for 𝑥 and
adding 52.5, we find that 𝐸𝐷 equals 84.
In conclusion, the length of
segment 𝐸𝐷 is 84, and the radius equals 52.5.
