Constructions & Loci

To bisect line segment $AB$ at right angles: set the compasses to more than half the length of $AB$. Draw arcs from $A$ above and below the line, then, with the same radius, arcs from $B$ crossing them.

Join the two crossing points with a straight line. It passes through the midpoint of $AB$ at $90^{\circ }$.

Every point on this line is the same distance from $A$ as from $B$.

To bisect an angle at vertex $V$: draw an arc centred at $V$ crossing both arms. From each crossing point, draw arcs of equal radius that meet inside the angle. Join $V$ to the meeting point.

Every point on the angle bisector is equidistant from the two arms.

To draw a perpendicular from a point $P$ to a line: draw an arc from $P$ cutting the line twice, then construct the perpendicular bisector of those two points; it passes through $P$.

Given three sides (SSS): draw one side accurately, then use compasses set to each of the other lengths from its ends; the arcs cross at the third vertex.

Given two sides and the included angle (SAS), or two angles and a side (ASA), use a protractor for the angles and a ruler for the sides.

Label vertices and leave all arcs visible.

Points a fixed distance from a point form a $\text{circle}$. Points a fixed distance from a line segment form a "running track" shape: two parallel lines joined by semicircles.

Points equidistant from two points lie on the $\text{perpendicular bisector}$. Points equidistant from two intersecting lines lie on the $\text{angle bisector}$.

"Closer to $A$ than $B$" means one side of the perpendicular bisector; "less than 3 m from $P$" means inside a circle. Combine conditions by finding where all regions overlap, and shade that region.