TopPerpendicular bisector to any line divides the line in two congruent parts. Let us know how to construct a perpendicular bisector of a triangle with a compass. Our Geometry would look like an Intersection of two lines making an angle of 900.
To start with, first draw a straight Line Segment PQ of suitable length using your ruler. Place the sharp end of the compass on any one end of the Line Segment say we place it on 'P'. Wide open the compass towards right side end Point i.e. 'Q'. It should go beyond mid - point at least. Using this measure of compass you have to draw one arc above the line and one arc below the line. Repeat the same procedure by putting compass’s sharp point on end point 'Q' this time. Wide open the compass beyond mid – point, this time towards the left.
Next draw two arcs again, one below the line and another above the line. Draw a line joining the intersection points of two arcs above the line and two below the line. Label the point of intersection of two arcs above the line Point 'R'. Label the point of intersection of two arcs below the line Point 'S'. So line segment joining points 'R' and 'S' points is a bisector of line segment PQ and also perpendicular to same line segment. This is how you draw a perpendicular bisector in a triangle where bisector passes through third point, other than two representing the line segment. So in a triangle we have two sub – triangles congruent to each other by property of SAS i.e. Side – Angle – Side property. Suppose we have a triangle whose perpendicular bisector is AD:
Here, triangles ADB and ADC are congruent to each other.