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How to build a bisector of a given angle? Building tasks

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The angle can be halved in the same way as a line segment. Divide in half - this means to divide something into two equal parts. There are two ways to divide the angle in half. You can use the protractor if it is and if you need to measure the angle. Or you can draw a bisector using a ruler and compass.

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.
  • Rearrange the compass needle in 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.
  • Do the same 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 "geometrical place of points". A bisector is the geometrical location of points that are equidistant from the rays that form an 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 an 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.

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