### Video Transcript

In the figure shown, the circle has
a radius of 12 centimeters, π΄π΅ equals 12 centimeters, and π΄πΆ equals 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.

The first thing we wanna do is take
information in our question and use it to label our diagram. We know the circle has a radius of
12 centimeters. That will be the distance from π
to any point on the outside of the circle. We also know that π΄π΅ is 12
centimeters and π΄πΆ equals 35 centimeters. Now, this also means we can find
out what the distance from π΅ to πΆ will be. And the distance from π΅ to πΆ must
be 35 minus 12, so itβs 23 centimeters. We want to find the distance from
π΄ to π·, and we want to find the distance from line segment π΅πΆ to the middle of
the circle. Weβre looking for the shortest
distance, and thatβs going to be the perpendicular distance from the line segment
π΅πΆ to the center of the circle π.

Since line segment π΄π· is a
tangent and it intersects the secant π΄πΆ at the point π΄, we can say that π΄π·
squared will be equal to π΄π΅ times π΄πΆ. And we know that π΄π΅ is 12
centimeters and π΄πΆ is 35 centimeters, which means π΄π΅ times π΄πΆ is 420
centimeters squared. But to find π΄π·, we need to take
the square root of both sides. Weβre only interested in the
positive square root, since weβre talking about distance. When we take the square root of
420, we get 20.4939 continuing. And we need to round that to the
nearest tenth. When we do that, we find π΄π·
equals 20.5 centimeters.

Now, finding the distance from line
segment π΅πΆ to π requires a little bit more information. We need to think about what we know
about triangles. We know the distance from π to πΆ
because from π to πΆ is a radius of this circle. We know that distance must be 12,
which means the distance from π to π΅ will also be 12 centimeters. What weβre now looking at is an
isosceles triangle, where the distance from π to the base is here. And the height of an isosceles
triangle is also its median, which means it divides the base in half. So, we can divide this 23
centimeters by two, which will give us this distance to be 11.5 centimeters.

And then, we can use the
Pythagorean theorem to find a missing side in a right-angled triangle. The Pythagorean theorem tells us
that π squared plus π squared equals π squared. In our case, the radius is the
hypotenuse, so we need to square 12. Weβll let π be our missing side
length and π be the 11.5 side length. So, we have π squared plus 11.5
squared equals 12 squared. π squared plus 132.25 equals
144. Then, we subtract 132.25 from both
sides, and we get π squared equals 11.75. Again, we need to take the square
root. We only want the positive solution
for distance. This gives us 3.4278 continuing,
which we round to 3.4. The distance from line segment π΅πΆ
to the center of the circle is 3.4 centimeters, and π΄π· equals 20.5
centimeters.