To check a construction, go back to the original wording and test the finished answer against each condition one at a time. A drawing can meet one rule by accident and quietly break another.
This is the final lesson in constructions and loci. It pulls together perpendicular bisectors, angle bisectors, single loci and combined loci.
What is a good checking routine?
- Copy the conditions into a short list, one line each, before you draw.
- Draw and label, keeping every construction arc.
- Test each condition separately, with a ruler or protractor.
- Calculate where you can, to see if the numbers agree with the drawing.
- Re-read the question for the exact word: exactly, within, closer to, at least.
Worked example
A and B are 10 cm apart. A buoy P must be equidistant from A and B and 13 cm from A. Check the student’s construction of P.
Step 1, list the conditions: (a) PA = PB, (b) PA = 13 cm.
Step 2, measure: PA measures 13.0 cm and PB measures 13.1 cm. Both are within 2 mm of 13, so (a) and (b) hold within tolerance.
Step 3, calculate: the midpoint M of AB is 5 cm from A. So MP² = 13² − 5² = 169 − 25 = 144 and MP = 12 cm.
Step 4, compare: the student measures MP as 12.0 cm. The drawing and the calculation agree.
Answer: P is 12 cm from AB, on the perpendicular bisector, and both conditions are met.
Double check: 5² + 12² = 25 + 144 = 169 = 13². This is the 5, 12, 13 right-angled triangle, so the calculation is right.
The mistake to watch for
A common slip is to check only the condition that was easiest to see.
Mistaken check: “PA is 13 cm, so the construction is correct.”
The student never measured PB, so there is no evidence that P is equidistant from A and B.
The correction is to list every condition before drawing, then tick each one after measuring. If PB had come out at 13.6 cm, the compass width would have drifted and the bisector would need redrawing.
Check yourself
Try each question, then open the answer.
1. AB = 16 cm. P is equidistant from A and B, with PA = 17 cm. Find the distance from P to AB.
Show answer
AM = 8 cm. MP² = 17² − 8² = 289 − 64 = 225, so MP = 15 cm. Check: 8² + 15² = 64 + 225 = 289.
2. A student measures PA = 8.0 cm and PB = 8.6 cm when P should be equidistant from A and B. Which condition fails and what should be done?
Show answer
The equidistant condition fails, because the two distances differ by 6 mm. Redraw the two arcs with one unchanged compass radius.
3. The region must be within 4 cm of A. A test point measures 4.3 cm from A. Is it in the region?
Show answer
No. 4.3 cm is more than 4 cm, so the point lies outside the circle of radius 4 cm. Move it or choose another point.
Where this leads next
Put the whole module together with the constructions and loci practice set. Use the non-calculator working trainer to check the arithmetic in your Pythagoras steps.
Some students find the drawing easy and the checking habit hard to keep under time pressure. Our teachers help build that habit in online one-to-one Mathematics tuition.