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Some basic constructions such as bisecting an angle and drawing a perpendicular bisector 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.
Discussion
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
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2
Draw Another Circle
expand_more 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
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Connect the points using a straightedge.
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.
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.
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
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2
Draw the Perpendicular Bisector of the Diameter
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Next, draw a segment perpendicular to at point This will be another diameter of Identify its endpoints as and
3
Connect the Points
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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
expand_more Given a circle, begin by drawing an arbitrary diameter. Then identify the endpoints of the diameter as and
2
Construct a Perpendicular Segment
expand_more 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
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4
Draw a Circle and Line
expand_more 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
expand_more 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
expand_more 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
expand_more 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
expand_more 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
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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.
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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