Loci

Fixed Distance from a Point: The locus of points at a constant distance $r$ from a fixed point $P$ is a circle with center $P$ and radius $r$. This represents all points that are exactly $r$ units away from the center.

Equidistant from Two Points: The locus of points that are the same distance from point $A$ and point $B$ is the perpendicular bisector of the line segment $AB$. This line passes through the midpoint of $AB$ at a 90-degree angle.

Equidistant from Two Intersecting Lines: The locus of points that are the same distance from two lines that meet at a vertex is the angle bisector. This line divides the angle between the two original lines into two equal parts.

Fixed Distance from a Line: The locus of points at a constant distance from a straight line consists of two parallel lines, one on each side of the original line. If the line is a segment, the locus also includes semi-circles at each end, forming a 'stadium' shape.

Points vs. Lines: A common confusion is between the perpendicular bisector (used for two points) and the angle bisector (used for two lines). Always identify if the 'targets' are points or lines before starting.

Equality vs. Inequality: A rule stating 'equal to' results in a line or curve, whereas 'less than' or 'more than' results in a shaded region. Boundary lines should be solid for $\le$ or $\ge$ and dashed for $<$ or $>$.

Leave Construction Arcs: Never erase your arc marks. Examiners look for these 'footprints' to verify that you used a compass and straightedge rather than just measuring with a ruler.

Accuracy is Vital: Ensure your pencil is sharp and your compass is tight. A deviation of even 1-2 millimeters can lead to incorrect intersections and lost marks in high-precision questions.

Check the Scale: Always verify the scale of the map or diagram (e.g., 1 cm = 2 m). Convert all real-world distances into drawing distances before setting your compass.

Shading Regions: If asked to shade a region satisfying multiple conditions, use small 'ticks' in areas that satisfy individual rules. The final shaded area should be the one where all conditions overlap.

The 'Halfway' Error: When constructing a perpendicular bisector, if the compass radius is set to exactly half or less than half the line length, the arcs will not intersect. Always set the radius to be 'visibly more than half'.

End-of-Line Loci: When drawing a locus a fixed distance from a line segment, students often forget the curved 'caps' at the ends. The locus is not just two parallel lines; it must curve around the endpoints to maintain a constant distance.

Vertex Confusion: In angle bisector problems, ensure the initial arc intersects both lines at the same distance from the vertex. If the intersection points are uneven, the resulting bisector will be skewed.