Rectangle 𝐵𝐶𝑀𝐷 is drawn inside
a quarter-circle, where 𝐵𝐷 equals nine centimeters and 𝐵𝐶 equals 12
centimeters. Find the length of arc 𝐴𝐵𝐸,
giving the answer in terms of 𝜋.
So, we’ve been asked to find the
length of the arc 𝐴𝐵𝐸. We know that we have quarter of a
circle, which means that this arc length will be quarter of the circle’s full
circumference. So, it’s equal to one-quarter
multiplied by 𝜋𝑑. 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, we’ve been given some
other measurements in the question. 𝐵𝐷 is nine centimeters and 𝐵𝐶
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 𝑀𝐴, 𝑀𝐵, and 𝑀𝐸. Now, as 𝐵𝐶𝑀𝐷 is a rectangle, we
know that the length of the line segment 𝐶𝑀 will be the same as the length of the
line segment 𝐵𝐷. So, it’s also nine centimeters.
If we look carefully at the figure,
we now see that we have a right triangle, triangle 𝐵𝐶𝑀, in which we know the
lengths of two of the sides; they’re 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, 𝑎 and
𝑏, is equal to the square of the hypotenuse. So, this is often written as 𝑎
squared plus 𝑏 squared equals 𝑐 squared.
In our triangle, the hypotenuse is
the radius of the circle because it’s the side directly opposite the right
angle. So, we have 𝑎 and 𝑏 as nine and
12 centimetres and 𝑐 as 𝑟 centimeters. We can, therefore, form an
equation, 𝑟 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,
that’s 144 plus 81, is 225. 𝑟 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, it’s 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; it’s 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 𝜋𝑑. We have one-quarter multiplied by
30𝜋. And we can then simplify by
canceling a factor of two from the numerator and denominator. We’re 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 𝐴𝐵𝐸 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.