Video Transcript
The circumference of a circle with
a center 𝑀 is 259 and the line segment 𝑍𝑌 is tangent at 𝑌. Calculate the length of the line
segment 𝑍𝑌 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 we’re told is
that the circumference of the circle is 259. Well, we know that there’s a
formula for the circumference of a circle. And this is that 𝑐 is equal to
𝜋𝑑. But how is this gonna be
useful? Well, if we take a look at our
sketch, we can see that 𝑋𝑌 is gonna be the diameter and that’s because we know
that the center is 𝑀. 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 𝜋𝑑. Well, this is going to give us that
259 divided by 𝜋 is equal to 𝑑. So that’s 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 we’ve got 𝑋𝑌 — our diameter — marked on. But is there anything else we can
do? Well, we’re told that the line
segment 𝑍𝑌 is tangent at 𝑌. 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 I’ve shown here. Once again, we know that 𝑀𝑌 is a
radius because we know that 𝑀 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 we’re trying to do is find the length of the line segment 𝑍𝑌. 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
we’ve got an angle. So, therefore, because we have
these things and because we’ve got this angle, what we’re gonna use is our
trigonometric ratios to help us solve this problem and find the line segment
𝑍𝑌. Well, the first thing we do if
we’re gonna use trigonometry is label our triangle.
So we’ve got the hypotenuse, which
is the longest side opposite the right angle. Then, we’ve got the opposite, which
is opposite the angle that we’ve been given. And then we’ve got the adjacent,
which is between the right angle and the angle we’ve been given. And also, it’s adjacent to the
angle we’ve been given. Well, whenever we’re solving a
trigonometry problem, we’ve got a stepped approach. Our first step was label, which
we’ve done. And the second step is to decide
which one of our trigonometric ratios we’re 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 that’s the line segment 𝑍𝑌. So therefore, if we look at our
SOHCAHTOA, we can see the 𝑂 and 𝐴, so the opposite and the adjacent, are part of
the tangent ratio. So, therefore, we’re gonna use tan
𝜃 is equal to the opposite divided by the adjacent.
So now, step three or the next step
is to substitute in the values we’ve got. So when we do that, what we’re
gonna get is that tan 30 is equal to 259 over 𝜋 over 𝐴, which is our adjacent. So now, what we’re gonna do to
rearrange and make 𝐴 the subject is multiply by 𝐴 and then divide by tan 30. So now, what we’re gonna do is put
it in the calculator because we’ve got the 𝐴, which is our line segment 𝑍𝑌, is
equal to 259 over 𝜋 divided by tan 30. So, therefore, we know that the
adjacent 𝐴 is gonna be equal to 142.79418 etc. But we haven’t finished here cause
what we’re looking for is the answer to the nearest hundredth.
Well, if we’re looking to the
nearest hundredth, it’s gonna be the second decimal place because the first decimal
place is the tenth. So, we’ve drawn a line here. And then, the number to the right
of this is gonna be the deciding number. And because it’s less than five,
then what we’re gonna do is round down. So, therefore, what we can say is
that the length of the line segment 𝑍𝑌 is 142.79 and that’s to two decimal places
or the nearest hundredth.
