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.
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 has the same length as AB.
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.
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.
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!
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, and have the same radius. Examine the ray with a starting point A and passing through the intersection points of and
One of the great things about constructions is that some constructions can even be used to produce other constructions.
The parallel line can be constructed in several steps.
Begin by drawing a line that intersects at any arbitrary location, and goes through the point. The point where the new line intersects will be named Q.
A copy of the angle formed by and will be constructed in order to transfer the copy to point P. To copy the angle, begin by fixing the needle of the compass at Q. With the compass set to an arbitrary length, draw an arc on between Q and P. Then, with the same setting, draw an arc from P across
Adjust the radius of the compass so that both compass ends are on the first drawn arc's intersection points with and Using that setting, place the needle of the compass where the second arc intersects Draw an arc intersecting the second arc. The point where these arcs intersect can be labeled R.
Maya and Tearrik are playing a construction game that they call Construct It if You Can! Tearrik starts the game by drawing AB and BC as shown. It is then Maya's turn to complete the drawing into a trapezoid. There is one catch to the game rules: Maya is only allowed to use copies of AB to complete the trapezoid.
It seems that AB looks shorter than BC and Maya can only use copies of AB. Therefore, one of the sides that will be drawn should be parallel to BC. Otherwise, the segments will not connect in the shape of a trapezoid. After the parallel line is drawn, the fourth side of the trapezoid can be drawn.
To draw a segment parallel to BC and through A, there are three steps to follow.
To make the next steps easier, AB ought to be extended using a straightedge.
Here, a copy of ∠B will be constructed in such a way that the copy will be located at point A. To do so, draw an arc centered at B across the angle. Then, with the same setting, draw another arc centered at A.
Adjust the compass so that both ends align with points of intersection of the first arc in relation to the two lines. With that compass setting, place the compass needle where the second arc intersects and draw an arc that intersects the second arc. Name the arcs' point of intersection R.
Finally, with the help of a straightedge, D and C can be connected. After deleting the auxiliary arcs and lines, a trapezoid can be seen.
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!