### Video Transcript

In the figure shown, the circle has
a radius of 12 centimeters. π΄π΅ is equal to 12 centimeters,
and π΄πΆ is equal to 35 centimeters. Determine the distance from line
segment π΅πΆ to the center of the circle π and the length of line segment π΄π·,
rounding your answers to the nearest tenth.

Weβre going to begin by finding the
distance from line segment π΅πΆ to the center of the circle π. We might recall that the shortest
distance from a point to a line is the length of the perpendicular from that point
to the line. And so we construct this
perpendicular from point π to the line segment π΅πΆ. In fact, since π is the center of
the circle and π΅πΆ is a chord, we can say that this perpendicular is the
perpendicular line bisector of π΅πΆ. So defining the point where this
perpendicular meets the line segment π΅πΆ as πΈ, we can say that π΅πΈ must be equal
to πΈπΆ.

Next, weβre going to use the fact
that the radius of the circle is 12 centimeters. The radius, of course, is the line
segment that joins the point of the center of the circle to any point on its the
circumference. So we can say that ππ΅ is 12
centimeters.

Next, we apply the fact that π΄π΅
is equal to 12 centimeters and π΄πΆ is equal to 35 centimeters. Since we can think of line segment
π΄πΆ as the sum of line segments π΄π΅ and π΅πΆ, we can say that 35 is equal to 12
plus π΅πΆ and we can find the length of π΅πΆ by subtracting 12 from both sides of
this equation. 35 minus 12 is 23. So π΅πΆ is 23 centimeters in
length.

But remember, we said that the line
segment ππΈ is the perpendicular bisector for the line segment π΅πΆ. So π΅πΈ must be half of π΅πΆ, that
is, 23 divided by two or 23 over two centimeters. We now note that we have a right
triangle ππΈπ΅ for which we know two of its sides. We can therefore use the
Pythagorean theorem to find the length of the side ππΈ. Letβs call that π₯ or π₯
centimeters.

Substituting what we know about
this triangle into the Pythagorean theorem, and we find that 12 squared equals π₯
squared plus 23 over two squared. Then we make π₯ squared the subject
by subtracting 23 over two squared from both sides. 12 squared minus 23 over two
squared is 47 over four. To find the length that weβre
interested in, π₯, weβre going to find the positive square root of 47 over four. And thatβs equal to 3.427 and so
on. Correct the nearest tenth, we find
thatβs equal to 3.4 centimeters.

We now move on to the second part
of this question. And that asks us to find the length
of line segment π΄π·. And we observed that line segment
π΄π· is, in fact, a tangent segment, whilst the line π΄πΆ is a secant segment. This means we can use a special
version of the intersecting secants theorem. And thatβs called the tangent
secant theorem. In the case of our circle, it tells
us that the product of the lengths of line segments π΄π΅ and π΄πΆ is equal to the
square of the length of line segment π΄π·.

Now weβre given that π΄π΅ is 12
centimeters, whilst π΄πΆ is 35. So 12 times 35 is equal to π΄π·
squared, or π΄π· squared is equal to 420. Weβll solve this equation by
finding the square root of 420. That gives us that π΄π· is 20.493,
which correct the nearest tenth is 20.5 centimeters. The distance from line segment π΅πΆ
to the center of the circle π is 3.4 centimeters, and the length of line segment
π΄π· is 20.5 centimeters.