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Explore

Using Circles to Construct Lines

Consider a segment with endpoints and There are two circles with the same radius — centered at and respectively. The radius can be adjusted, and a line that passes through the two circles' points of intersection can be drawn.
Two circles and segment joining their centers
How is and the line passing through the circles' points of intersection related? There are several basic constructions in Geometry.

Discussion

Constructing a Copy of a Segment

Throughout the lesson, the constructions mentioned above will be performed. It is best to start with the most straightforward construction, that is, copying a segment.

Discussion

Constructing a Bisector of a Segment

Now, it will be shown how to construct the geometric objects that bisect a segment.

Discussion

Constructing the Perpendicular Bisector of a Segment

The steps taken to bisect a segment are also used to construct a perpendicular bisector.

When each arc in this construction is completed into a circle, the graph in the exploration slide is obtained. Therefore, the line drawn in the exploration slide is the perpendicular bisector of

Example

Solving Problems by Constructing Bisectors

Constructions, such as drawing a line, can come in handy when building things in real life. For example, in carpentry, it is common practice to divide wooden material in half.

Vincenzo likes to practice wood carving in his free time. He is given a basswood log, from his abuelita. The log has already been cut into a square shape. To make a carving he has in mind, he wants to reshape the log into a rectangular shape with a height that is twice its width.

Top view of a log

Help Vincenzo draw a line that divides the log into a rectangle and meets the given characteristics. He would then be able to use this line to cut the wood to obtain the shape he wants.

Answer

See solution.

Hint

Choose one side and draw its perpendicular bisector.

Solution

The perpendicular bisector of one side will divide the log in half, creating two rectangles.
Dividing a square
To draw the perpendicular bisector, place the tip of the compass on one of the vertices of the square log. Just by eyeballing, open the compass beyond the midpoint of and draw an arc. Afterwards, place the tip on the other adjacent endpoint and draw another arc with the same radius.
Drawing arcs on the log

Repeat this procedure using a larger width than the previous one.

Drawing arcs on the log

Finally, using a straightedge, draw the line that connects the intersection points of the arcs.

Connecting the points of intersection of the arcs

As can be seen, with the help of a straightedge and a compass, the perpendicular bisector has been drawn. By cutting the wood with the help of this line, Vincenzo can get the rectangular piece with a height that is twice its width. He even ends up with a second piece!

Discussion

Constructing a Perpendicular Line

Explore

Using Circles to Construct Angles

In the second half of the lesson, the other basic constructions that involve angles will be understood. Consider the angle The applet shows three circles on the angle. Two of the circles, and have the same radius. Examine the ray with a starting point and passing through the intersection points of and

Angle A and circles A, B, and C
What can be concluded about the ray and the angle? Is it possible to find a segment that is perpendicular to the ray?

Discussion

Constructing a Copy of an Angle

Discussion

Constructing the Bisector of an Angle

An angle bisector consists of points that are equidistant from the two sides of the angle, creating a ray. The applet on the previous explore slide illustrates those points and the ray.

Discussion

Constructing Parallel Lines

One of the great things about constructions is that some constructions can even be used to produce other constructions.

Closure

Constructions and Real Life Applications

Considering the constructions studied throughout the lesson, it is noteworthy that the compass is used to set equidistant points and the straightedge is used to set collinearity.

Geometric constructions are practical not only for showing geometric relationships or suggesting new ones but also for making designs. For example, a flower with six petals can be drawn by a compass alone. Give it a try!
6 petal flower design
Designers might consult constructions when creating company logos. For example, the Apple logo can be constructed using a compass and a straightedge. Take a look at some other designs and think about how they were made using a construction.