How do you construct perpendicular bisectors of a triangle?

Question

Caroline Aucoin · Accepted Answer

Draw two circles of the the same radius equal to the length of given segment #AB# with centers at #A# and #B#. They intersect at two points #P# and #Q#. Line #PQ# is a perpendicular bisector to #AB#.

To construct perpendicular bisectors of a triangle #Delta ABC# you have to consider each side separately as a segment (#AB#, #BC# and #AC#) and construct a perpendicular bisector to each of them.

The easy way to construct a perpendicular bisector #PQ# to segment #AB# is pictured below.

Here the centers of these circles are the endpoints of a given segment #AB# and their radiuses must be the same. The only condition is for these circles is the existence two points of intersection, #P# and #Q#. For this the radius can be any, as long as it's greater than half of the length of #AB#. The simple method is to choose it to be equal to the length of #AB#.

What's more interesting is to prove that this construction delivers the perpendicular bisector.
Here is the proof.

Assume that #M# is an intersection of #AB# and #PQ#.
#AP=BP=AQ=BQ# - each is a radius, which we have chosen
#Delta APQ = Delta BPQ# - by side-side-side theorem

Hence:
#=>/_APQ=/_BPQ# as angles of congruent triangles lying across congruent sides #AQ# and #BQ#
#=> Delta APM = Delta BPM# by side-angle-side theorem
#=> AM=BM# as sides of congruent triangles lying across congruent angles #/_APM=/_BPM#
#=>/_AMP=/_BMP# as angles of congruent triangles lying across congruent sides #AP# and #BP#
#=>/_AMP=/_BMP=90^o# since their sum is #180^o#

So, we have proven that #M# is a modpoint of #AB# and #PM_|_AB#.

Sadie Allen · Answer

undefined To construct perpendicular bisectors of a triangle, follow these steps:

1. Draw the triangle using a ruler and pencil.
2. For each side of the triangle, use a compass to draw arcs that are more than halfway across the side from each endpoint.
3. Repeat step 2 for the other two sides of the triangle.
4. Where the arcs intersect on each side, draw a point. These points are the midpoints of the sides.
5. Use a straightedge to connect each midpoint to the opposite vertex of the triangle.
6. The lines drawn in step 5 are the perpendicular bisectors of the triangle. They intersect at the circumcenter, which is equidistant from each vertex of the triangle.