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Here are a few recommended readings before getting started with this lesson.
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.
Using a straightedge and a compass, it is possible to construct a copy of a segment.
To copy AB, these three steps can be followed.To make a segment with the same length as AB, measure the original segment with a compass. To do so, place the needle point on one of its endpoints and place the pencil on the other end.
It has been shown that A′B′ has the same length as AB.
Now, it will be shown how to construct the geometric objects that bisect a segment.
To bisect a segment means to draw the object that bisects it, or passes through its midpoint.
Here, a bisector of AB will be constructed using a compass and a straightedge.Keep the compass set to the same length. Then place the needle point at the other end of the segment and draw a second arc across the segment.
If the arcs have been drawn large enough, they will now intersect each other twice. If they don't, extend them.
The steps taken to bisect a segment are also used to construct a perpendicular bisector.
On any given segment, a compass and a straightedge can be used to draw its perpendicular bisector.
To draw the perpendicular bisector of a segment, these three steps can be followed.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.
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.
See solution.
Choose one side and draw its perpendicular bisector.
Repeat this procedure using a larger width than the previous one.
Finally, using a straightedge, draw the line that connects the intersection points 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!
A line perpendicular to a given line through a point can be drawn by using a compass and straightedge. This perpendicular line is unique and is described by the Perpendicular Postulate.
Given a line and a point, a perpendicular line through the point can be drawn by following these steps.
In the second half of the lesson, the other basic constructions that involve angles will be understood. Consider the angle BAC. The applet shows three circles on the angle. Two of the circles, ⊙B and ⊙C, have the same radius. Examine the ray with a starting point A and passing through the intersection points of ⊙B and ⊙C.
An angle can be constructed as a copy of a given angle using a compass and a straightedge.
The given angle can be copied in four steps.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.
An angle can be bisected using a compass and a straightedge.
To bisect an angle, these three steps can be followed.
One of the great things about constructions is that some constructions can even be used to produce other constructions.
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!