### Video Transcript

The circumference of a circle with
a center ๐ is 259 and the line segment ๐๐ is tangent at ๐. Calculate the length of the line
segment ๐๐ to the nearest hundredth.

Well, the first thing we do with
this type of question is mark on any values we know. Well, the first thing weโre told is
that the circumference of the circle is 259. Well, we know that thereโs a
formula for the circumference of a circle. And this is that ๐ is equal to
๐๐. But how is this gonna be
useful? Well, if we take a look at our
sketch, we can see that ๐๐ is gonna be the diameter and thatโs because we know
that the center is ๐. So, therefore, what we can do is
use our formula and the information we know to work out the diameter. Because what we can say is that 259
is gonna be equal to ๐๐. Well, this is going to give us that
259 divided by ๐ is equal to ๐. So thatโs our diameter.

Well, what we could have done now
is calculated what this is. But we leave this in this form for
accuracy because it prevents any rounding issues later in the problem. Okay, so we have this side marked
on, or weโve got ๐๐ โ our diameter โ marked on. But is there anything else we can
do? Well, weโre told that the line
segment ๐๐ is tangent at ๐. But what does this mean? Well, we know that if a line is
tangent to a circle at a point, then at this point a right angle is gonna be formed
between the tangent and the radius, which Iโve shown here. Once again, we know that ๐๐ is a
radius because we know that ๐ is the center of the circle.

Okay, great. So, now, what do we need to do? Well, in the question, we can see
that what weโre trying to do is find the length of the line segment ๐๐. Well, what we, in fact, have now is
a right triangle, where we have one side, then a side we want to find, and then
weโve got an angle. So, therefore, because we have
these things and because weโve got this angle, what weโre gonna use is our
trigonometric ratios to help us solve this problem and find the line segment
๐๐. Well, the first thing we do if
weโre gonna use trigonometry is label our triangle.

So weโve got the hypotenuse, which
is the longest side opposite the right angle. Then, weโve got the opposite, which
is opposite the angle that weโve been given. And then weโve got the adjacent,
which is between the right angle and the angle weโve been given. And also, itโs adjacent to the
angle weโve been given. Well, whenever weโre solving a
trigonometry problem, weโve got a stepped approach. Our first step was label, which
weโve done. And the second step is to decide
which one of our trigonometric ratios weโre going to use.

And to help us do that, what we
have is a memory aid, which is SOHCAHTOA. And what this tells us is that sine
of an angle is equal to the opposite divided by the hypotenuse. The cosine of an angle is equal to
the adjacent divided by the hypotenuse. And the tangent of an angle is
equal to the opposite divided by adjacent. Well, if we take a look at our
problem, we can see that we know the opposite and we want to find the adjacent
because thatโs the line segment ๐๐. So therefore, if we look at our
SOHCAHTOA, we can see the ๐ and ๐ด, so the opposite and the adjacent, are part of
the tangent ratio. So, therefore, weโre gonna use tan
๐ is equal to the opposite divided by the adjacent.

So now, step three or the next step
is to substitute in the values weโve got. So when we do that, what weโre
gonna get is that tan 30 is equal to 259 over ๐ over ๐ด, which is our adjacent. So now, what weโre gonna do to
rearrange and make ๐ด the subject is multiply by ๐ด and then divide by tan 30. So now, what weโre gonna do is put
it in the calculator because weโve got the ๐ด, which is our line segment ๐๐, is
equal to 259 over ๐ divided by tan 30. So, therefore, we know that the
adjacent ๐ด is gonna be equal to 142.79418 etc. But we havenโt finished here cause
what weโre looking for is the answer to the nearest hundredth.

Well, if weโre looking to the
nearest hundredth, itโs gonna be the second decimal place because the first decimal
place is the tenth. So, weโve drawn a line here. And then, the number to the right
of this is gonna be the deciding number. And because itโs less than five,
then what weโre gonna do is round down. So, therefore, what we can say is
that the length of the line segment ๐๐ is 142.79 and thatโs to two decimal places
or the nearest hundredth.