{{ stepNode.name }}

{{ 'ml-toc-proceed' | message }}

An error ocurred, try again later!

Chapter {{ article.chapter.number }}

{{ article.number }}. # {{ article.displayTitle }}

{{ article.intro.summary }}

{{ 'ml-btn-show-less' | message }} {{ 'ml-btn-show-more' | message }} {{ 'ml-heading-abilities-covered' | message }}

{{ 'ml-heading-lesson-settings' | message }}

{{ 'ml-lesson-show-solutions' | message }}

{{ 'ml-lesson-show-hints' | message }}

| {{ 'ml-lesson-number-slides' | message : article.intro.bblockCount}} |

| {{ 'ml-lesson-number-exercises' | message : article.intro.exerciseCount}} |

| {{ 'ml-lesson-time-estimation' | message }} |

Consider a segment with endpoints $A$ and $B.$ There are two circles with the same radius — centered at $A$ and $B,$ respectively. The radius can be adjusted, and a line that passes through the two circles' points of intersection can be drawn.

How is $AB$ and the line passing through the circles' points of intersection related? There are several basic constructions in Geometry.

- Copying a segment
- Copying an angle
- Bisecting a Segment / Drawing a perpendicular bisector
- Bisecting an angle
- Drawing a perpendicular line through a given point
- Drawing a line parallel to a given line and through a given point

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.

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

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.

The perpendicular bisector of one side will divide the log in half, creating two rectangles.

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 $AB,$ and draw an arc. Afterwards, place the tip on the other adjacent endpoint and draw another arc with the same radius.

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!

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 $⊙C.$

What can be concluded about the ray and the angle? Is it possible to find a segment that is perpendicular to the ray?

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.

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! 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.