To combine two loci, draw each rule as its own shape on the same diagram. The positions that satisfy both rules lie where the shapes meet or overlap.
This lesson follows describing a locus using distance conditions in constructions and loci, and it uses the perpendicular bisector from the first lesson.
How do you approach a two-rule question?
Take one rule at a time. Write each one as a locus, draw it, and only then look at the overlap. Exactly gives a line or curve, so the answer is usually a point.
Within or closer to gives a region, so the answer is an area.
The phrase “closer to A than to B” means one side of the perpendicular bisector of AB, and it is the side containing A.
Worked example
A rectangular garden ABCD has AB = 8 m and AD = 6 m. A lamp post is to stand at a point P that is equidistant from A and B and exactly 5 m from A. Find P.
Step 1, first locus: equidistant from A and B means P lies on the perpendicular bisector of AB. It meets AB at the midpoint M, 4 m from A.
Step 2, second locus: exactly 5 m from A means P lies on a circle of radius 5 m centred at A.
Step 3, find the crossing: the crossing point P forms a right-angled triangle AMP with AM = 4 and AP = 5. So MP² = 25 − 16 = 9 and MP = 3.
Step 4, inside the garden: P is 3 m above M. The other crossing is 3 m below AB, which is outside the garden. The garden is 6 m wide, so 3 m fits.
Answer: P is on the perpendicular bisector of AB, 3 m from AB, inside the garden.
Check: PA = √(4² + 3²) = 5 and PB = 5. Both conditions hold.
The mistake to watch for
A common slip is to read a region word as a line.
Mistaken answer: for “closer to A than to B, and within 5 m of A”, the student draws only the circle and the bisector, then marks one point.
The question asks for a region, not a point.
The correction is to shade the part of the circle on A’s side of the bisector. Test a point such as M shifted slightly towards A: it is within 5 m of A and closer to A than to B, so the shading is right.
Check yourself
Use the garden from the example, then open each answer.
1. Find the point P that is equidistant from A and B and 6 m from A, in the garden where AB = 8 m. Give the distance from AB to 1 decimal place.
Show answer
AM = 4 and AP = 6, so MP² = 36 − 16 = 20 and MP = √20 ≈ 4.5 m. It is inside the garden because 4.5 is less than 6.
2. Is there a point that is equidistant from A and B and 3 m from A when AB = 8 m?
Show answer
No. Points on the bisector are at least AM = 4 m from A. A point 3 m from A never reaches the bisector.
3. With AB = 8 m, find the length along AB of the points that are within 5 m of A and within 5 m of B.
Show answer
Within 5 m of A reaches 5 m along AB. Within 5 m of B reaches back to 8 − 5 = 3 m from A. The overlap runs from 3 m to 5 m, which is 2 m.
Where this leads next
Once you have a candidate answer, you need to test it against the original wording, which is the subject of checking a construction against the original constraints. The non-calculator working trainer helps with the square-root steps.
Some students draw the two loci well but are unsure which region is required. That is the kind of decision our teachers practise in online one-to-one Mathematics tuition.