175. The Perfect Split · Cutting a segment exactly in half at a right angle
The bisector cuts BC exactly in half and stands at 90 degrees.
What this lesson covers
Try to break it
Drag B and C apart or together. The bisector always passes through the exact midpoint of BC and stands at 90° to it. Try to drag B and C into a position where the bisector tilts off 90° or misses the midpoint — you can't. Both follow B and C automatically.
How you build it
Construct the perpendicular bisector of BC.
- Draw the line segment BC using the line tool.
- Place the compass point at B and draw an arc with radius greater than half of BC.
- Without changing the radius, place the compass at C and draw another arc to intersect the first one at P and Q.
- Draw a line through points P and Q using the line tool. This is your perpendicular bisector.
The proof, step by step
Prove that the constructed line bisects BC at right angles.
- BP = CP (By construction, radii of equal arcs)
- BQ = CQ (By construction, radii of equal arcs)
- Hence, P and Q are equidistant from B and C.
- The line joining two points each equidistant from the ends of a line segment is the perpendicular bisector of the segment. So, PQ is the perpendicular bisector of BC.
Worked example
In the construction of a perpendicular bisector of a line segment, why is the radius of the arcs taken more than half the length of the segment?
The radius must be greater than half the segment length to ensure the arcs drawn from both endpoints intersect each other. Without intersection, we cannot locate points P and Q to draw the bisector.
- To make the drawing look bigger
- So the arcs intersect each other — correct
- To ensure the compass is sharp
- It does not matter what the radius is