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Challenge

Positioning Three Cameras in a Circular Room

Jordan is the head of security at a famous casino hotel named The Compazu in Las Vegas. She decided to have a safe room constructed after a heist took place. About its design, the safe room will be circular, and three security cameras will monitor it. One of the cameras will be opposite the entrance door.

Jordan wants to place the other two cameras equidistant from each other. Given the blueprint of the safe room, a compass, and a straightedge, how can she determine the positions of the other two cameras?

Explore

Investigating an Inscribed Square in a Circle

Given a square and a circle, use the applet to place the circle such that the square is inscribed in the circle. Then draw the diagonals of the square.

What can be concluded about the relationship between the location of the center of the circle and the point of intersection of the diagonals of the square?

Discussion

Inscribing a Square in a Circle

Considering the previous exploration, when given a circle, the construction of a square can be done using a compass and a straightedge.

Discussion

Inscribing a Regular Pentagon in a Circle

Discussion

Inscribing a Regular Hexagon in a Circle

None of the previous constructions will help Jordan. Finally, by using a compass and a straightedge, a regular hexagon will be constructed hoping that it will help Jordan. Otherwise, she may lose her job.

Closure

Solving the Problem of the Three Cameras

Among all of the constructions covered in this lesson, there is one in particular that will help Jordan place the cameras correctly — slowing any attempted heist! Recall that Jordan wants to determine the positions of the three cameras, given that one of the cameras will be opposite the entrance door.

Blueprint of the room

She came ready with a compass, a straightedge, and the blueprint of the room. Show how she can determine the positions of the other two cameras.

Answer

See solution.

Hint

Follow the process of inscribing a regular hexagon in a circle.

Solution


Following the process of inscribing a regular hexagon in a circle, first determine six points on the circle. The starting point will be the position of the first security camera.

According to the construction, each point on the circle is equidistant from each other. Therefore, it can be concluded that the distances between every other point are also equal.

Therefore, Jordan can determine the position of the cameras by choosing every other point starting from the position of the first point — Camera 1.

With this camera setup, Jordan can have three security cameras equidistant from each other, adequately monitoring The Compazu's safe room.


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