Constructions: Perpendicular and Angle Bisectors

1. 1 Explain the rules: compass and straight edge only — no measuring for constructions (though we measure to check afterwards). 'A construction is an exact geometric proof, not an approximation.'
2. 2 Draw line AB (about 8cm). Open compass to more than half the line length. Place on A, draw arc above and below the line. Without changing compass width, place on B, draw intersecting arcs. Draw a straight line through the two intersections. 'This line cuts AB exactly in half and is at 90°.'
3. 3 Students construct a perpendicular bisector of their own 10cm line. Check by measuring both halves (should both be 5cm) and using a set square to verify the 90° angle.
4. 4 Draw angle ABC. Place compass on vertex B, draw arc cutting both arms. Place compass on each intersection point, draw two arcs that cross inside the angle. Draw line from B through the intersection. 'This line divides the angle exactly in half.'
5. 5 Students draw a 70° angle (use protractor), construct the bisector, then check by measuring each half (should both be 35°). Repeat with a 100° angle.
6. 6 A gardener wants to plant a tree equidistant from two paths that meet at a corner. Construct the angle bisector to find the line of equidistant points. A post must be equidistant from two fence posts — construct the perpendicular bisector to find valid positions.
7. 7 Exchange work. Check: are the arcs visible and matching? Does the bisector look accurate? Measure to verify within 1mm and 1° tolerance.

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