BTT Mathematics / Primary Mathematics Learning Hub / Loci and Distance Constraints
A locus is the set of all points satisfying a stated geometric condition. A distance constraint may produce a line, circle, strip, pair of lines or a bounded region. The task is not to find one convenient point; it is to describe every permitted point and no others.
All points exactly three centimetres from a fixed point P form a circle centred at P with radius three centimetres. All points equidistant from two fixed points A and B lie on the perpendicular bisector of AB. These are geometric descriptions of solution sets.
This guide is Primary Mathematics enrichment. It connects Geometry, Angles, Symmetry and Coordinates, Scale Drawings and Maps, and Inequalities and Feasible Ranges. Formal locus construction is optional extension; use the MOE Primary curriculum page and current school scope for required content.
Distance from a point · Distance from a line · Equidistant loci · Coordinate-grid constraints · Combined conditions · 24 questions · Worked answers · Teaching and transfer
1. Fixed distance from one point gives a circle
Every point exactly r units from fixed point P lies on a circle of radius r centred at P. Points closer than r lie inside; points farther lie outside.
Worked example A: Exactly five units
The locus of points exactly five units from P is a circle centre P, radius 5 units.
Worked example B: At most five units
The set of points at most five units from P is the closed disk: the circle and all points inside it. The boundary is included because “at most” includes equality.
Worked example C: Less than five units
“Less than five” gives the interior of the disk but excludes the boundary circle.
Worked example D: Between two radii
Points with distance at least three and at most six from P form an annular region between circles radius three and six, including both boundary circles.
2. Fixed perpendicular distance from a line gives parallel lines
Distance from a point to a line is measured along the shortest perpendicular segment.
Worked example E: Exactly two units from line l
The locus consists of two lines parallel to l, one on each side, each at perpendicular distance two.
Worked example F: At most two units from line l
The locus is the strip between those two parallel lines, including both boundary lines.
Worked example G: One-sided condition
If the condition additionally says “on the north side of l,” only the parallel line or half-strip on that side remains.
Orientation conditions can remove a symmetric second solution and should not be ignored.
3. Equidistant from two points gives a perpendicular bisector
Let A and B be distinct points. Every point equidistant from A and B lies on the line perpendicular to AB through its midpoint.
Worked example H: Horizontal segment
A=(0,0), B=(8,0). The midpoint is (4,0), so the equidistant locus is the vertical line x=4.
Worked example I: Vertical segment
A=(2,−3), B=(2,5). Midpoint y=1, so the perpendicular bisector is horizontal line y=1.
Worked example J: Which side is closer?
For A=(0,0), B=(8,0), points with x<4 are closer to A than B; points with x>4 are closer to B. The line x=4 is the equality boundary.
Equidistant from two intersecting lines
For two intersecting lines, the points equidistant from the lines lie on their angle bisectors. This is optional extension and depends on perpendicular distance to each line.
4. Coordinate-grid constraints can be read as geometric regions
On a coordinate grid, x and y restrictions create vertical and horizontal boundaries. For example, 2≤x≤5 gives a vertical strip between x=2 and x=5.
Worked example K: Rectangle from four inequalities
1≤x≤4 and 2≤y≤6 describe a rectangle including its boundary. Its width is three and height four, so area=12 square units.
Worked example L: Whole-number grid points
For integer coordinates with 0≤x≤2 and 1≤y≤3, there are three possible x-values and three y-values, giving 9 grid points.
Worked example M: Distance along a horizontal line
On y=2, points exactly four horizontal units from (3,2) are (−1,2) and (7,2).
Worked example N: Axis constraints
Points on the y-axis satisfy x=0. Points on the x-axis satisfy y=0. Their intersection is the origin.
Optional grid metric
Some puzzles define distance as number of horizontal and vertical unit steps rather than straight-line distance. Under that rule, points exactly two steps from the origin form a diamond-shaped set of grid points such as (2,0),(1,1),(0,2) and their symmetric counterparts. The metric must be stated explicitly because it differs from ordinary Euclidean distance.
5. Combined loci are intersections of spatial conditions
When a point must satisfy two conditions, draw or describe each locus and keep only their overlap.
Worked example O: Circle and line
Points exactly five units from the origin and on the x-axis satisfy x²=25 and y=0, giving (−5,0) and (5,0).
Worked example P: Perpendicular bisector plus x-axis
A=(0,0), B=(6,0). The equidistant locus is x=3. Requiring the point also to lie on the x-axis leaves the single point (3,0).
Worked example Q: Two circles
Two circles may meet in two points, one point or no points. A combined distance condition can therefore have two, one or zero solutions.
Worked example R: Feasible rectangular region
A delivery point must satisfy 0≤x≤6 and 1≤y≤4. If only integer grid points are allowed, there are seven x-values and four y-values, giving 28 feasible grid points.
Worked example S: Nearest of two facilities
With facilities A=(0,0) and B=(8,0), the perpendicular bisector x=4 divides the plane into two regions. The side x<4 is closer to A, x>4 closer to B, and x=4 is tied. This is a geometric classification boundary.
6. Practice: 24 original questions
Unless a question explicitly defines grid-step distance, use ordinary straight-line geometric distance.
Questions 1–8: Standard loci
1. Describe the locus of points exactly 5 cm from fixed point P.
2. Describe the locus of points at most 5 cm from P.
3. Describe the locus of points exactly 2 cm from a straight line l.
4. Describe the locus of points at most 2 cm from line l.
5. A=(0,0), B=(8,0). Write the equation of the locus equidistant from A and B.
6. A=(2,−3), B=(2,5). Write the equation of the locus equidistant from A and B.
7. For A=(0,0), B=(8,0), which side of x=4 contains points closer to A?
8. Describe points whose distance from P is at least 3 and at most 6 units.
Questions 9–16: Grid regions and points
9. Describe the region 2≤x≤5.
10. Find the area of the rectangle 1≤x≤4, 2≤y≤6.
11. For integer coordinates 0≤x≤2 and 1≤y≤3, how many grid points are possible?
12. On line y=2, find the two points exactly 4 horizontal units from (3,2).
13. What equation describes the y-axis?
14. What equation describes the x-axis?
15. Where do the x-axis and y-axis meet?
16. A point has integer coordinates with 0≤x≤6 and 1≤y≤4. How many feasible grid points are there?
Questions 17–24: Combined distance constraints
17. Find points on the x-axis exactly 5 units from the origin.
18. A=(0,0), B=(6,0). Find the point that is equidistant from A and B and also lies on the x-axis.
19. With A=(0,0), B=(8,0), is point (3,5) closer to A or B?
20. With A=(0,0), B=(8,0), is point (4,100) closer to A, closer to B or equidistant?
21. A point must satisfy 1≤x≤3 and 4≤y≤5 with integer coordinates. List the number of possible points.
22. Under grid-step distance, list the four axis points exactly 2 steps from the origin.
23. Can two distance-circle conditions have no common solution? Answer yes or no and explain geometrically.
24. A point must be within 4 units of P and also exactly 6 units from P. Is any point feasible?
7. Worked answers
Answers 1–8
1. A circle centred at P with radius 5 cm.
2. The closed disk of radius 5 cm centred at P.
3. Two lines parallel to l, one on each side, each 2 cm away.
4. The closed strip between those two parallel lines.
5. x=4. It is the perpendicular bisector of AB.
6. y=1. The midpoint is (2,1), and the perpendicular bisector is horizontal.
7. x<4. That half-plane is closer to A.
8. The annular region from radius 3 through radius 6, including both boundaries.
Answers 9–16
9. The closed vertical strip between x=2 and x=5.
10. 12 square units. Width 3, height 4.
11. 9 points. Three x-values times three y-values.
12. (−1,2) and (7,2).
13. x=0.
14. y=0.
15. (0,0).
16. 28 points. Seven x-values times four y-values.
Answers 17–24
17. (−5,0) and (5,0).
18. (3,0). The perpendicular bisector is x=3.
19. Closer to A. Point (3,5) lies left of the equidistance boundary x=4.
20. Equidistant. Every point on x=4 is equally distant from A and B.
21. 6 points. Three x-values and two y-values.
22. (2,0), (−2,0), (0,2), (0,−2).
23. Yes. Two circles can be separate with the distance between centres greater than the sum of their radii.
24. No. A point cannot be both at distance at most four and exactly six from the same centre.
8. Teaching and transfer
If a learner marks one point and stops, ask for another point satisfying the same condition, then another. The repeated examples should suggest the full locus rather than a single solution.
When “distance from a line” is measured diagonally
Draw several segments from one point to a line. The perpendicular one is shortest. The locus at fixed line-distance is therefore parallel to the original line, not a curve created from arbitrary slanted measurements.
When two-point equidistance is guessed from the midpoint only
The midpoint is one equidistant point, but not the whole locus. Move perpendicularly away from the midpoint and compare the two symmetric distances. This reveals the entire perpendicular bisector.
When inequalities are separated from geometry
Translate x≤4 into a half-plane or 1≤x≤3 into a strip. The Inequalities guide and this locus guide describe the same solution-set idea in one and two dimensions.
When scale affects loci
On a map, convert the real distance to drawing distance before constructing the locus. A 2 km radius does not mean a 2 cm circle unless the scale says so.
Continue through this enrichment collection
For numerical feasible regions, use Inequalities, Number Lines and Feasible Ranges. For rule transformations, use Function Machines, Input–Output Tables and Inverse Rules. For step-by-step spatial decisions, use Mathematical Algorithms, Flowcharts and Decision Trees.
Return to the BTT Primary Mathematics Learning Hub.
Original enrichment guide with 24 original practice questions and separate worked answers. Distance conventions are stated locally; no surveying or real-world positioning data are implied.