L2 · Constructing bisectors

Learning goal

“I can state and order the steps for constructing a perpendicular bisector of a segment and an angle bisector, and I can use what a bisector guarantees: it passes through the midpoint at 90° (segment) or splits the angle into two equal angles, and every point on a perpendicular bisector is the same distance from both endpoints.”

Bisect — to cut exactly in half: a bisector produces two equal parts.

Perpendicular bisector — the line that crosses a segment at its midpoint AND at 90°. It is built with the same equal-arc method as L1's perpendicular.

Midpoint — the point exactly halfway along a segment, the same distance from both endpoints.

Equidistant — the same distance away. Every point on a perpendicular bisector is equidistant from the segment's two endpoints.

Worked example

Construct the perpendicular bisector of segment AB, length 10 cm. Step 1: Open the compass to MORE than half of AB - more than 5 cm, say about 7 cm. (Why more: the arcs from the two ends must be able to REACH each other and cross.) Step 2: Point at A: draw an arc above AB and another below AB, same opening. Step 3: WITHOUT changing the opening, point at B: draw arcs that cross the first pair - one crossing above (call it Q) and one below (call it R). Step 4: Draw the line through Q and R. That line is the perpendicular bisector: it crosses AB at its midpoint (5 cm from A) and at exactly 90°. Say it plainly: L1's perpendicular needed arcs that cross ONCE; a bisector of a whole segment needs crossings on BOTH sides so the straightedge has two points to join.

Check yourself — explain it

The opening must be more than half the segment because...

Paired problem — example still visible

Segment CD is 8 cm. To start its perpendicular bisector, the compass opening must be more than ___ cm. -> 4 cm

Faded completion — last step is yours

"Perpendicular bisector: 1) opening more than half; 2) arcs from both endpoints, above and below; 3) the two crossing points are joined by ___." -> A straight line (the perpendicular bisector itself)

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