Question

Explain how to bisect AB, that is, to divide AB

into two congruent line segments. Then bisect
AB and use the result to find the midpoint M
of AB.

Answer 1

To bisect a line segment AB, draw arcs from points A and B with the same radius that intersect and then draw the perpendicular bisector through the intersection points. This line will cross AB at the midpoint M, dividing it into two congruent segments.

Step-by-step explanation:

To bisect a line segment AB and find its midpoint M, one can follow these general geometric construction steps, which rely on using a compass and a straightedge:

1. Place the compass at one end of line segment AB (point A), and draw an arc above and below the line. The radius of the arc should be greater than half the length of AB.
2. Without changing the radius, repeat step 1 from the other end of the line segment (point B), creating two more arcs that intersect with the first set.
3. The intersections of these arcs create two new points on opposite sides of AB. Draw a straight line through these points; this line is the perpendicular bisector of AB and crosses AB at its midpoint, M.
4. The point where the perpendicular bisector intersects AB is the midpoint M, dividing AB into two congruent line segments, AM and MB.

This construction ensures that segments AM and MB are of equal length, hence congruent, since a perpendicular bisector of a segment divides it into two equal parts.