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Congruence, Proof, and Constructions

Polygons Inscribed in a Circle

In the previous lessons, some basic constructions were made using several tools and methods. With those constructions as a foundation, this lesson will expand into more complex constructions such as constructing an equilateral triangle and a regular pentagon using a straightedge and a compass.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

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

Constructing an Equilateral Triangle With a Compass

An equilateral triangle can be constructed in several ways. One method is to use two intersecting circles.

An equilateral triangle is a triangle with three congruent sides.

It can be constructed using a compass and a straightedge.

1

Draw a Circle

Begin by drawing a circle centered at point Then identify an arbitrary point on the circumference.


2

Draw Another Circle

Place the compass at point B and draw another circle passing through point A. Then identify one of the intersection points of the circles as point


3

Connect the Points


Connect the points using a straightedge.

An equilateral triangle has been constructed.


Why

Note that point and point are the centers of and respectively. Since passes through point and passes through point the radii of the circles are equal. Therefore, and are equal. Consequently, is an equilateral triangle.

Unfortunately, this construction is not convenient for Jordan to place the cameras. Jordan needs to find another alternative way. Some other methods will be searched throughout the course.

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.

A square that is inscribed in a circle can be constructed in a few of steps.

Given a circle, these three steps can be followed to construct a square inscribed in the circle.

1

Draw a Diameter


Begin by drawing an arbitrary diameter of using a straightedge. Identify the endpoints as and


2

Draw the Perpendicular Bisector of the Diameter


Next, draw a segment perpendicular to at point This will be another diameter of Identify its endpoints as and


3

Connect the Points


Finally, by connecting the endpoints of the diameters, an inscribed square can be drawn.


Why

Note that diagonals of a square are congruent and they are the perpendicular bisectors of each other. Since and are congruent and they are perpendicular bisectors of each other, polygon is a square.

Discussion

Inscribing a Regular Pentagon in a Circle

A regular pentagon can be inscribed in a circle using a compass and straightedge.

The following steps can be followed to construct a regular pentagon.

1

Draw a Diameter

Given a circle, begin by drawing an arbitrary diameter. Then identify the endpoints of the diameter as and


2

Construct a Perpendicular Segment

Construct a perpendicular segment to at point After that identify the point of intersection of the circle and the perpendicular segment as


3

Construct the Perpendicular Bisector

Construct the perpendicular bisector of and identify the intersection point as


4

Draw a Circle and Line

Next, draw a circle centered at point with a radius of Then draw a line passing through point and point intersecting From here, identify the intersection points of and the segment as and


5

Draw the Arcs

Now, two arcs centered at point will be drawn. Both arcs will intersect One of them will pass through point and the other one will pass through point Label the intersection points of the arcs and as and


6

Connect the Points

Finally, connect points and


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.


These three steps can be followed to construct a regular hexagon inscribed in a circle.

1

Draw a Circle

Given place the compass on an arbitrary point on the circumference of the circle. Then draw a circle passing through the center of and identify one of the intersection points as


2

Repeat the Process to Obtain Six Intercepted Points on the Circle

Next, place the compass at point and draw another circle passing through the center of Repeat the process until is intercepted six times and label the intersection points as and


3

Connect the Points


Finally, connect the points using a straightedge.

Therefore, a regular hexagon has been constructed.


Why


Notice that a regular hexagon consists of six equilateral triangles that are congruent.

By repeating the process of constructing an equilateral triangle several times, the vertices of the polygon has been found. Therefore, the resulting polygon is a regular hexagon.


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

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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