Loci — chapter notes

• A locus is the set of all the points, and only those points, that obey a given rule: a curve, a line, a pair of lines or a region.
• Fixed distance from a fixed point $O$: a circle with centre $O$ and that distance as radius.
• Inside: $OX < r$; on the circle: $OP = r$; outside: $OY > r$. With coordinates, compare $x^2 + y^2$ with $r^2$.
• Drawing at a scale: a real radius $R$ drawn at $1 : n$ has radius $\dfrac{R}{n}$ on the plan.

Example: the tip of a $7$ cm minute hand is always $7$ cm from the pivot: a circle of radius $7$ cm.

• The points at a fixed distance $d$ from a line $l$ form two lines parallel to $l$, one on each side, each at distance $d$.
• Distance from a line is measured at right angles to the line.
• The two parallels are $2d$ apart; if only one side is allowed, only one parallel is the locus.

Example: a coin of radius $1.5$ cm rolling along an edge has its centre on a line parallel to the edge and $1.5$ cm from it.

• The points equally far from $A$ and $B$ lie on the perpendicular bisector of $AB$: it passes through the mid-point $M$ at right angles to $AB$.
• Construction: arcs of equal radius (more than $\dfrac{AB}{2}$) from $A$ and from $B$ meet at $X$ and $Y$; join $XY$.
• A point $P$ on the bisector has $PM = \sqrt{PA^2 - AM^2}$.
• The proofs of the locus theorems are not in the ICSE 2027 syllabus: the results are used.

Example: $AB = 6$, $PA = 5$: $PM = \sqrt{25 - 9} = 4$ cm.

• The points equally far from two lines that cross lie on the bisectors of the angles between them: two lines.
• The two bisectors are at right angles: $\dfrac{\theta}{2} + \dfrac{180^\circ - \theta}{2} = 90^\circ$.
• Construction: arcs from the vertex cut the arms at $P$, $Q$; equal arcs from $P$ and $Q$ meet at $R$; join the vertex to $R$.
• A point on a bisector at distance $s$ from the vertex is $s \sin\left(\dfrac{\theta}{2}\right)$ from each arm.

Example: an angle of $70^\circ$, $P$ at $4$ cm on its bisector: distance $4 \sin 35^\circ \approx 2.3$ cm.

• The points equally far from two parallel lines lie on one line parallel to both, midway between them.
• If the lines are $g$ apart, the middle line is $\dfrac{g}{2}$ from each.
• Two parallels have no crossing point, so there is just one locus line (unlike crossing lines, which give two).

Example: edges $3.6$ m apart: the line of lamps is $1.8$ m from each edge.

• A point that obeys two conditions lies on both loci: it is where they intersect.
• A line and a circle meet in 2, 1 or 0 points (secant, tangent, no contact); two circles meet in $2$ points if $|r_1 - r_2| < d < r_1 + r_2$, in $1$ point if $d = r_1 + r_2$ (or $|r_1 - r_2|$), and in none otherwise ($d$ is the distance between centres).
• Always ask: how many solutions are there? Does each one satisfy every condition (for example, inside the triangle)?

Example: $A$, $B$ $4$ cm apart; points $3$ cm from $A$ and equidistant from $A$ and $B$: $2$ points, $PQ \approx 4.5$ cm.

• Plan: (1) construct the given figure, (2) draw each locus, (3) mark the point where they meet, (4) measure and answer with units.
• Keep all construction arcs; use a sharp pencil; label every point.
• Typical pairs: bisector of an angle with the perpendicular bisector of a side; bisector with a line parallel to a side; a circle with a perpendicular bisector; two perpendicular bisectors (the point equidistant from three corners).
• Regions: "nearer to $B$ than to $C$" is one side of the perpendicular bisector of $BC$; "within $r$ of $A$" is inside a circle.

Example: triangle $BC = 6.5$, $\angle B = 60^\circ$, $AB = 5$: the bisector of $B$ meets the perpendicular bisector of $BC$ at $P$ with $BP \approx 3.8$ cm.