Loci and Perpendicular Distance

Distance to a line means perpendicular distance

Imagine a point $P$ beside a straight line. You could join $P$ to many different points on the line, but the shortest route meets the line at a right angle. This is the perpendicular distance from $P$ to the line. A sloping measurement is longer and does not give the distance to the line. To construct the shortest route accurately, use the ruler-and-compass perpendicular from $P$ to the line.

A locus collects every possible position

A locus is all the points that satisfy a rule. For example, every point exactly $3\text{ cm}$ from a fixed point $A$ lies on a circle centred at $A$ with radius $3\text{ cm}$. The circle is the whole locus, not just one point on it. Set a compass to $3\text{ cm}$ and draw the circle to construct it.

Other distance rules lead to lines. Points equally distant from two fixed points $A$ and $B$ lie on the perpendicular bisector of $AB$: it passes through the midpoint of $AB$ at $90°$. Points exactly $3\text{ cm}$ from a straight line lie on two parallel lines, one on each side. Construct a perpendicular to measure $3\text{ cm}$ from the given line, then draw a parallel through each marked point. The measured distance stays $3\text{ cm}$ all along either parallel line.

Points equally distant from two intersecting lines lie on their angle bisectors. If the problem restricts the points to one angle, use the bisector within that angle. Here, too, distance to each line is measured perpendicularly.

Combine conditions to solve a loci problem

Suppose $A$ and $B$ are $6\text{ cm}$ apart. Find positions that are equally distant from $A$ and $B$ and exactly $4\text{ cm}$ from $A$. First construct the perpendicular bisector of $AB$ to represent the equal-distance condition. Then draw a circle of radius $4\text{ cm}$ centred at $A$. Their two intersections are the possible positions: each is on the perpendicular bisector and on the circle. A point elsewhere on either construction satisfies only one rule.

Words such as within or no more than describe a region rather than just its edge. Within $4\text{ cm}$ of $A$ means the inside of the $4\text{ cm}$ circle, including its boundary. Within $2\text{ cm}$ of a straight line means the strip between two parallels $2\text{ cm}$ from the line, including the boundaries. When several rules are given, keep only the points or part of the region common to all of them.

Distance and loci

Distance from a point to a line is measured perpendicularly; this is the shortest distance.

• Fixed distance from a point → circle.
• Equal distances from two points → perpendicular bisector of the line joining them.
• Fixed distance from a straight line → two parallels, one on each side.
• Equal distances from intersecting lines → angle bisectors.

Combining rules

Construct a locus for each condition. The answer is their intersection or overlapping region. Exactly gives a boundary; within includes the inside and its boundary.