Geometric Properties and Constructions

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infoWhy this? Compass-and-straightedge constructions create exact geometric objects from defining properties rather than measurement alone. Bisectors and perpendiculars develop precision, locus reasoning, and an understanding of equidistance.

scheduleWhy now? Earlier geometry established perpendicular lines, angle properties, congruent constructions, and accurate instrument use. This final Year 9 unit formalises those foundations through standard constructions that support loci, scale drawings, and geometric proof.

A perpendicular bisector is a line that meets a line segment at `90^\circ` and passes through its midpoint.
Every point on the perpendicular bisector of a line segment is the same distance from the segment’s two endpoints.
If a point is the same distance from the two endpoints of a line segment, that point lies on the perpendicular bisector of the segment.
An angle bisector divides an angle into two equal angles.
Every point on an angle bisector is the same perpendicular distance from the two arms of the angle.
A line perpendicular to another line meets it at a right angle of `90^\circ`.
A straightedge is used to draw straight lines in constructions, and a compass is used to draw arcs and circles.
Accurate geometric constructions are done with a compass and straightedge, not by measuring with a ruler or a protractor.
To construct a perpendicular bisector, the compass width must be more than half the length of the line segment so that arcs from each endpoint intersect.
When constructing a perpendicular bisector, the two intersecting arc points are each equidistant from the segment’s endpoints, so the line through them is the perpendicular bisector.
To bisect an angle, an arc from the vertex marks one point on each arm, and equal-radius arcs from those two points create an intersection on the angle bisector.
To construct a perpendicular through a point on a line, equal distances are marked on either side of the point on the line so the required line is the perpendicular bisector of the new segment.
To construct a perpendicular from a point not on a line, a compass arc centred at the point can cut the line in two places, creating a segment whose perpendicular bisector passes through the original point.
Construction marks such as arcs should be left visible because they show the geometric reasoning used to locate the required line.

rocket_launchYou must be able to

Construct the perpendicular bisector of a line segment by drawing equal-radius arcs from each endpoint, with radius greater than half the segment, and joining the arc intersections with a straight line.
Bisect a given angle by drawing an arc from the vertex to cut both arms, drawing equal-radius arcs from those cut points, and joining the vertex to the new arc intersection.
Construct a perpendicular line through a point on a line by marking equal distances on both sides of the point, drawing intersecting arcs from those two marks, and joining the given point to the arc intersection.
Construct a perpendicular line from a point not on a line by drawing an arc centred at the point to cut the line twice, drawing equal-radius arcs from those two cut points, and joining the external point to the arc intersection.
Set and maintain a fixed compass width accurately during each stage of a construction so that pairs of arcs are equal where required.
Check a completed construction by verifying the correct property, such as equal angle parts, a midpoint with a right angle, or equal distances from the constructed line.
Present constructions clearly by using a sharp pencil, drawing precise arcs, and leaving construction lines visible.

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