![]() ![]() We also know that DC=BC since they are both radii of the circle with center C and radius BC. We know that EC is still perpendicular because we know DE=BE as they are both legs of an equilateral triangle and EDC=EBC because they are both angles of an equilateral triangle. We still construct an equilateral triangle DBE. In this case, we already know the center point, but that doesn’t change our procedure much. Now we can proceed as if we were constructing a perpendicular bisector on the segment DB. Then, label the intersection of this circle and the line AB as D. We can do this by constructing a circle with a radius equal to the shorter of AC and BC. To do this, we first have to create a line segment that has C at its center. Example 2Ĭonstruct a line perpendicular to the given line at point C. Therefore, the triangles ACE and BCE are congruent, as are the sides AE and BE. This is because AC=BC, ACE=BCE, and CE are equal to themselves. We know AE and BE are equal in length because the triangles ACE and BCE are congruent. Finally, if we connect CD and label the intersection of CD and AB as E, we will have found the center of AB. ![]() Then, we can construct a second equilateral triangle by connecting A and B to the other intersection of the circles, D. If we construct lines from A and B to the intersection of the circles, C, we will construct an equilateral triangle ABC. The first will have center A, and the second will have center B. Example 1įind the center of the given line segment.įirst, we construct an equilateral triangle on the line segment AB by creating two circles with radius AB. In this section, we will go over common example problems involving the construction of perpendicular bisectors. This is true for any triangle, scalene, isosceles, or equilateral. The intersection of these three bisectors will be the circumcenter. To do this, we must construct a perpendicular bisector for each of the three legs of the triangle and draw it the whole way through the center of the triangle. That is, we use them to find a point inside a triangle that is equidistant from each of the vertices. Perpendicular bisectors are useful for finding the circumcenter of a triangle. ANSWERS TO GSP5 CONSTRUCTING PERPENDICULAR BISECTORS HOW TOHow to Construct the Perpendicular Bisector of a Triangle Therefore, CE is also perpendicular to AB. Since the two resulting angles, CEA and CEB, are congruent and adjacent, they are right angles. Thus, E is the center of the segment AB, and CE bisects AB. This means that the third sides, namely AE and BE, are equivalent. Therefore, since the triangles, ACE and BCE, have two sides the same and the angle between those sides the same, the two triangles are congruent. We can first prove that E is the center of AB by showing that AE=BE.ĪC=BC because they are both legs of an equilateral triangle, ACE=BCE because CE bisects ACB, and CE is equal to itself. Call the intersection of the angle bisector and the line AB E. Then, we must bisect the angle ACB (how-to here). Label the intersection of these circles as C and draw segments AC and BC. The first will have center A, while the second will have center B. We want to construct a line that meets this segment at a right angle and divides the given segment into two equal parts.įirst, we draw two circles with length AB. How to Construct a Perpendicular Bisector of a Given Line Segment We can then prove that this line will meet the given line at its center and form a right angle. Then, we extend the angle bisector so that it intersects the initial line. How to Construct a Perpendicular BisectorĪ perpendicular bisector is a line that meets a given line segment at a right angle and cuts the given line segment into two equal halves.Ĭonstructing such a line requires that we draw an equilateral triangle on the given line segment and then bisect the third vertex. ![]() How to Construct the Perpendicular Bisector of a Triangle.How to Construct a Perpendicular Bisector of a Given Line Segment.How to Construct a Perpendicular Bisector.To do this requires constructing an equilateral triangle on the line segment.īefore moving on, review the construction of a perpendicular line. Constructing Perpendicular Bisectors – Explanation and ExamplesĬonstructing a perpendicular bisector with a compass and straightedge requires that we first find the center of a line segment and then construct a line perpendicular to that point. ![]()
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