### Video Transcript

In this video, we will learn how to
use the theory of the perpendicular bisector of a chord from the center of a circle
and its converse to solve problems. Letβs begin by recalling how we can
define a radius, a chord, and a diameter. A radius is a line segment which
has one end at the center of the circle and the other on the circumference. We define a chord as any straight
line segment whose endpoints both lie on the circumference of the same circle. The diameter is a special type of
chord which passes through the center of the circle. We can also consider this to be
made up of two radii.

We will now consider what a
perpendicular bisector of a chord looks like. In the diagram drawn, we have the
chord π΅πΆ together with its perpendicular bisector. We will look at three theorems in
this video. And in each case, we need to
consider the center of the circle π΄ together with the radii π΄π΅ and π΄πΆ.

Our first theorem states that if we
have a circle with center π΄ containing a chord π΅πΆ, then the straight line that
passes through π΄ and bisects the chord π΅πΆ is perpendicular to π΅πΆ. The second theorem is very
similar. If we have a circle with center π΄
containing a chord π΅πΆ, then the straight line that passes through π΄ and is
perpendicular to π΅πΆ also bisects π΅πΆ. Our third theorem is the converse
to the chord bisector theorem. This states that if we have a
circle with center π΄ containing a chord π΅πΆ, then the perpendicular bisector of
π΅πΆ passes through π΄.

It is important to note that the
perpendicular bisector of a chord creates two congruent triangles. In the diagram drawn, these are
triangles π΄π·πΆ and π΄π·π΅. We will now look at some examples
where we can use the theorems discussed to find missing lengths and angles.

Given π΄π equals 200 centimeters
and ππΆ equals 120 centimeters, find the length of the line segment π΄π΅.

In the diagram shown, we have a
circle with center π. The chord π΄π΅ is bisected by the
line segment ππ· at the point πΆ. Applying the chord bisector
theorem, which states that if we have a circle with center π containing a chord
π΄π΅, then the straight line that passes through π and bisects the chord π΄π΅ is
perpendicular to π΄π΅. We can say that the measure of
angle ππΆπ΅ is 90 degrees. We are told that the length of π΄π
is 200 centimeters. And since this is a radius of the
circle, π΅π is also 200 centimeters. We are also told that ππΆ is equal
to 120 centimeters. Adding these measurements to our
diagram, we have a right triangle ππΆπ΅ as shown.

Applying the Pythagorean theorem,
the length of π΅πΆ is equal to the square root of 200 squared minus 120 squared. This is equal to 160. As π΅πΆ is equal to 160 centimeters
and the point πΆ bisects the line segment π΄π΅, then π΄πΆ is also equal to 160
centimeters. The line segment π΄π΅ is therefore
equal to 320 centimeters.

In our next example, we will see
how we can apply the theroems to find the area of a triangle.

In the figure below, if ππ΄ is
equal to 17.2 centimeters and π΄π΅ is equal to 27.6 centimeters, find the length of
the line segment ππΆ and the area of triangle π΄π·π΅ to the nearest tenth.

Since π is the center of the
circle and the line segment ππ· bisects the chord π΄π΅ at πΆ, we can apply the
chord bisector theorem. This states that if we have a
circle with center π containing a chord π΄π΅, then the straight line that passes
through π and bisects π΄π΅ is perpendicular to π΄π΅. This means that the measure of
angle ππΆπ΄ is 90 degrees. Since the chord π΄π΅ has length
27.6 centimeters and we know that πΆ bisects this chord, π΄πΆ is equal to 27.6
divided by two. This is equal to 13.8
centimeters.

We are also told that the radius
ππ΄ is equal to 17.2 centimeters. We can therefore use the
Pythagorean theorem in the right triangle ππΆπ΄ such that ππΆ is equal to the
square root of 17.2 squared minus 13.8 squared. This is equal to 10.266 and so
on. And rounding to the nearest tenth,
the line segment ππΆ is equal to 10.3 centimeters.

The second part of our question
asks us to calculate the area of triangle π΄π·π΅. Clearing some space, we recall that
the area of any triangle is equal to the length of its base multiplied by the length
of its perpendicular height divided by two. We know that the base of our
triangle π΄π΅ is equal to 27.6 centimeters. The perpendicular height πΆπ· will
be equal to the length of ππ· minus the length of ππΆ. ππ· is the radius of the circle,
and we know this is equal to 17.2 centimeters. Whilst we could use 10.3
centimeters for ππΆ, it is better for accuracy to use the nonrounded version: ππΆ
is equal to 10.266 and so on. Subtracting this from 17.2, we see
that the length of πΆπ· is 6.933 and so on centimeters.

We can now calculate the area of
triangle π΄π·π΅ by multiplying this by 27.6 centimeters and then dividing by
two. This is equal to 95.683 and so
on. Once again, we need to round to the
nearest tenth, giving us an answer of 95.7 square centimeters.

In our next example, we will see
how we can use perpendicular bisectors of chords to find a missing angle.

Line segments π΄π΅ and π΄πΆ are two
chords in the circle with center π in two opposite sides of its center, where the
measure of angle π΅π΄πΆ is 33 degrees. If π· and πΈ are the midpoints of
the line segments π΄π΅ and π΄πΆ, respectively, find the measure of angle π·ππΈ.

We begin by noticing that ππΈ and
ππ· both pass through the center of the circle and that they bisect the chords π΄πΆ
and π΄π΅, respectively. We can therefore apply the chord
bisector theorem, which states if we have a circle with center π containing a chord
π΄π΅, then the straight line which passes through π and bisects π΄π΅ is
perpendicular to π΄π΅. This means that, on our diagram,
the measure of angle ππΈπ΄ and the measure of angle ππ·π΄ are both equal to 90
degrees.

We notice that π΄π·ππΈ is a
quadrilateral. And we know that the angles in a
quadrilateral sum to 360 degrees. This means that the measure of
angle π·ππΈ which we are trying to calculate is equal to 360 minus 90 minus 90
minus 33. This is equal to 147 degrees.

In our final example, we will find
the perimeter of a triangle using perpendicular bisectors of chords.

In a circle of center π, π΄π΅ is
equal to 35 centimeters, πΆπ΅ is equal to 25 centimeters, and π΄πΆ is equal to 40
centimeters. Given that line segment ππ· is
perpendicular to line segment π΅πΆ and line segment ππΈ is perpendicular to line
segment π΄πΆ, find the perimeter of triangle πΆπ·πΈ.

We are given in the question the
length of the three sides of the triangle πΆπ΅π΄. We know that π΄π΅ is 35
centimeters, πΆπ΅ is 25 centimeters, and π΄πΆ is 40 centimeters. We have been asked to calculate the
perimeter of triangle πΆπ·πΈ. We will do this by firstly proving
that triangles πΆπ΅π΄ and πΆπ·πΈ are similar using the chord bisector theorem. We notice from the diagram that the
line segments ππΈ and ππ· both pass through π and meet the chords π΄πΆ and πΆπ΅
at right angles.

The chord bisector theorem states
that if we have a circle with center π containing a chord π΅πΆ, then the straight
line that passes through π and is perpendicular to π΅πΆ also bisects π΅πΆ. In our diagram, this means that the
length of π΄πΈ is equal to the length πΈπΆ and the length πΆπ· is equal to the
length π·π΅.

It is also clear from the diagram
that π΄πΆ is equal to two multiplied by πΈπΆ and πΆπ΅ is equal to two multiplied by
πΆπ·. As the two triangles πΆπ΅π΄ and
πΆπ·πΈ also share the angle πΆ, we have two corresponding sides in proportion and
the angle between the two sides is congruent. This proves that the two triangles
are similar. And in fact triangle πΆπ΅π΄ is
larger than triangle πΆπ·πΈ by a scale factor of two, as the lengths of the
corresponding sides are twice as long. Side π΄πΆ is equal to two
multiplied by side πΈπΆ, πΆπ΅ is equal to two πΆπ·, and π΄π΅ is equal to two
multiplied by πΈπ·.

We can calculate the perimeter of
triangle πΆπ΅π΄ by adding 40, 35, and 25. This is equal to 100
centimeters. The perimeter of triangle πΆπ·πΈ
will therefore be equal to half of this. This is equal to 50
centimeters.

We have now seen a variety of
examples of how perpendicular bisectors of chords can be used to find missing
lengths, angle measures, and other unknowns in problems involving circles. We will now recap the key points
from this video.

The chord bisector theorem can be
summarized in three ways. Firstly, if we have a circle with
center π΄ containing a chord π΅πΆ, then the straight line that passes through π΄ and
bisects the chord π΅πΆ is perpendicular to π΅πΆ. In the same way, the straight line
that passes through π΄ and is perpendicular to π΅πΆ also bisects π΅πΆ. The converse of these two states
that the perpendicular bisector of the chord π΅πΆ passes through the center π΄. As already stated, these theorems
can be used to find missing lengths, angle measures, and other unknowns in problems
involving circles.