Bisecting a Line and an Angle
≈ 38 minBisecting a Line and an Angle
To bisect is to cut exactly in half using only compasses and a straight edge — no measuring, no protractor. This matters because a measured half carries your ruler's error, while a constructed half is exact by geometry.
Bisecting a line AB. Open the compasses to more than half of AB. Swing an arc from A, then — without changing the radius — an arc from B. The two arcs cross above and below the line. The line joining those crossings cuts AB in half and is perpendicular to it. You get two results from one construction.
Bisecting an angle. From the vertex, swing an arc cutting both arms. From each of those two cuts, swing equal arcs that cross. Join the vertex to that crossing.
Worked example. Why must the radius exceed half of AB? If it is smaller, the arcs never meet and there are no crossings to join. If it is exactly half, they touch at a single point on the line itself and cannot define a second point to draw through.
Core checkpoint: Both arcs must use the same radius. Changing the compass setting between them destroys the symmetry the construction depends on.
When bisecting line AB, why must the compass radius be more than half of AB?
A learner swings the arc from A, then widens the compasses slightly before swinging the arc from B. What is wrong with the result?
Besides cutting AB in half, what is the other property the bisector automatically has in relation to AB?
To bisect an angle, what is the correct first step after being given the two arms and the vertex?