### Video Transcript

Rectangle π΅πΆππ· is drawn inside
a quarter-circle, where π΅π· equals nine centimeters and π΅πΆ equals 12
centimeters. Find the length of arc π΄π΅πΈ,
giving the answer in terms of π.

So, weβve been asked to find the
length of the arc π΄π΅πΈ. We know that we have quarter of a
circle, which means that this arc length will be quarter of the circleβs full
circumference. So, itβs equal to one-quarter
multiplied by ππ. We can also think of that value of
one-quarter as 90 over 360 if we wish because this quarter-circle is also a sector
of a circle with a central angle of 90 degrees.

In order to answer this problem
then, we need to know the diameter or perhaps the radius of this circle. Instead, weβve been given some
other measurements in the question. π΅π· is nine centimeters and π΅πΆ
is 12 centimeters, but neither of these are the radius or diameter. We can identify three radii of this
circle in the diagram: the lines ππ΄, ππ΅, and ππΈ. Now, as π΅πΆππ· is a rectangle, we
know that the length of the line segment πΆπ will be the same as the length of the
line segment π΅π·. So, itβs also nine centimeters.

If we look carefully at the figure,
we now see that we have a right triangle, triangle π΅πΆπ, in which we know the
lengths of two of the sides; theyβre 12 centimeters and nine centimeters. And the third side is the radius of
the circle, which we wish to calculate. If we know two sides in a right
triangle and we want to calculate the third, we can do this using the Pythagorean
theorem, which tells us that the sum of the squares of the two shorter sides, π and
π, is equal to the square of the hypotenuse. So, this is often written as π
squared plus π squared equals π squared.

In our triangle, the hypotenuse is
the radius of the circle because itβs the side directly opposite the right
angle. So, we have π and π as nine and
12 centimetres and π as π centimeters. We can, therefore, form an
equation, π squared equals 12 squared plus nine squared. And we can solve this equation to
find the radius of the circle. 12 squared plus nine squared,
thatβs 144 plus 81, is 225. π is, therefore, equal to the
square root of 225, which is 15.

Now, you may actually have already
realized this because nine, 12, 15 is an example of a Pythagorean triple. In fact, itβs an enlargement of the
three, four, five Pythagorean triple with a scale factor of three. And the three, four, five triple is
probably the most easily recognizable. In any case, we now know the radius
of the circle; itβs 15 centimeters. We can, therefore, calculate the
diameter of the circle by doubling this value. The diameter is 30 centimeters.

So, all that remains then is to
substitute this diameter into our formula for the arc length, which remember was
one-quarter multiplied by ππ. We have one-quarter multiplied by
30π. And we can then simplify by
canceling a factor of two from the numerator and denominator. Weβre left with 15 over two π,
which we can write using decimals, if we wish, as 7.5π.

The question asks us to leave our
answer in terms of π. So, we can include the units of
centimeters and we have our answer to the problem. The length of arc π΄π΅πΈ is 7.5π
centimeters. Notice that at no point in this
question did we need a calculator. We left our answer in terms of
π. And in our work with the
Pythagorean theorem, we were working with a Pythagorean triple.