### Instruction: A delicious circular pie with diameter $12\text{ cm}$ is cut into three equal-sized sector-shaped pieces. Let $l$ be the number of centimeters in the length of the longest line segment that may be drawn in one of these pieces. What is $l^2$?### Response: I know that the length of the longest line segment in a sector is the diameter of the circle, which is $12\text{ cm}$.
So I could just say $l=12$ and $l^2=144$ and be done with it.
But that seems too easy.
Maybe there is a catch.
Maybe the problem means the longest line segment that is entirely contained in the sector, not crossing the arc or the radius.
In that case, $l$ would be smaller than $12$.
How can I find it?
I can draw a picture of one of the sectors and label some angles and lengths.
Since the pie is cut into three equal pieces, each sector has a central angle of $120^\circ$.
The radius is $6\text{ cm}$.
I can draw the longest line segment inside the sector as a perpendicular bisector of the arc, so it forms two right triangles with the radii.
Now I have a right triangle with a hypotenuse of $6\text{ cm}$ and an angle of $60^\circ$.
I can use the sine or cosine function to find the length of the opposite or adjacent side, which is also half of $l$.
I'll use the cosine function, since I remember that $\cos 60^\circ = \frac{1}{2}$.
So I have $\cos 60^\circ = \frac{\text{adjacent}}{6}$.
Solving for the adjacent side, I get $\text{adjacent} = 6 \cos 60^\circ = 6 \cdot \frac{1}{2} = 3\text{ cm}$.
This means that half of $l$ is $3\text{ cm}$, so $l$ is $6\text{ cm}$.