There is such a funny nursery rhyme with which it is easy to remember what a bisector is: "A bisector is such a rat that runs around the corners and divides the angle in half." However, we must not forget that, despite the simplicity of remembering this comic definition, the teacher rightly demands another, taken from the textbook.
In further studying the school curriculum, children are faced with a difficult task at first glance - how to build a bisector of a given angle using a compass. However, a more advanced student will easily cope with this task, which is the basis for the implementation of the cycle of tasks for building in geometry. Let’s deal with this issue once and for all.
How to build a bisector of a given angle?
The most obvious and easiest way is to use a protractor, but if this auxiliary tool was not at hand, you must be able to build a bisector without it.
To accomplish this task, as the reader already understood, we need a compass, and in addition to it, a ruler (it is important to understand that you cannot use divisions on it) and a simple pencil with an eraser.
Construction algorithm
It is necessary to perform the following actions:
- Set the compass needle at the top of this corner.
- Set the compasses to an arbitrary radius, rotate the tool so that the arc drawn by it intersects both rays that form an angle.
- Mark the points of intersection of the arc with the sides of the given angle.
- Move the compass needle into one of the marked points, select an arbitrary radius, and again rotate the compass so that the arc drawn by it is enclosed inside the corner.
- Perform the same steps by moving the compass to the point marked on the other side of the corner. It is important to keep the radius selected in the previous paragraph of the algorithm.
- Mark the intersection point of the two arcs that were drawn in the previous two points.
- Draw a ray from the top of the corner passing through this point.
- The resulting beam is the desired one.
We answered the question posed - how to build a bisector of a given angle.
Evidence
Now, having figured out how to build a bisector of a given angle, it is worth recalling another definition of a bisector using the term "geometric location of points". The bisector is the locus of points that are equidistant from the rays that make up the angle.
According to the construction performed in paragraphs 4-6, the point belonging to the constructed bisector also belongs to two circles of equal radius, the center of which is located on the rays forming an angle at the same distance from the top of the corner (according to points 1-3 of the construction). We drop the perpendicular from the point noted in point 6 to the rays forming the angle. Let us prove that the resulting right-angled triangles are equal, and find out that the omitted perpendiculars are also equal as the corresponding elements of the triangles. Thus, their general hypotenuse is the angle bisector by definition. Q.E.D.