A construction is correct because each step forces a geometric condition. To explain it, name the condition each step creates.
This lesson is part of geometric construction reasoning. The next lesson, comparing construction steps in two diagrams, applies the same habit to a pair of diagrams.
What does “explain using constraints” mean?
Replace “I did this” with “this makes that true”. A step list says what happened. A constraint explanation says why the final shape has to be right.
Three constraints cover most school-level constructions: equal distances (same compass radius), perpendicular lines (right angles), and tangency (a line touching a circle at one point, at right angles to the radius).
Worked example: a perpendicular bisector
Given a line AB of length 60 mm, construct its perpendicular bisector.
- Set the compass to 40 mm, more than half of AB. Draw arcs above and below the line from A.
- Keep the compass at 40 mm. Draw arcs from B that cut the first arcs at P and Q.
- Join P and Q. This line cuts AB at M.
Explanation, step by step:
- Steps 1 and 2 use the same radius, so P is 40 mm from A and 40 mm from B. The same is true of Q.
- A point equally far from A and B lies on the perpendicular bisector of AB, so P and Q both lie on it.
- Two points fix one line, so PQ is the perpendicular bisector, and M is the midpoint with AM = MB = 30 mm.
The radius had to be more than 30 mm. If it were 25 mm, the arcs from A and B would never meet.
Worked example: a circle tangent to a line
A line L is given with a point T on it. Construct a circle of radius 25 mm that touches L at T.
- Draw a perpendicular to L at T.
- Mark O on the perpendicular, 25 mm from T.
- With centre O and radius 25 mm, draw the circle.
The radius OT is perpendicular to L by step 1, and a line at right angles to a radius at its end touches the circle at one point only. So L is tangent at T. The centre is forced onto the perpendicular, because any other position would make OT slant and cut L twice.
The mistake: explaining by appearance
A student writes, “PQ is the middle because it looks like it passes through the centre of AB.” This describes the drawing, not the geometry, and earns no reasoning marks.
| Explanation | Wrong | Right |
|---|---|---|
| Basis | The line looks central | P and Q are each 40 mm from A and B |
| Conclusion | Appears to be the middle | They lie on the perpendicular bisector |
| If the drawing is resized | The line may no longer look central | Still true at every size |
A constraint-based explanation survives any change of size. An appearance-based one does not.
Check yourself
A point P is to be placed 50 mm from A and 50 mm from B, where AB = 70 mm. Explain why P lies on the perpendicular bisector of AB, and say how far P is from the midpoint M of AB.
Answer
P is equally far from A and B (both 50 mm), so it lies on the perpendicular bisector of AB. M is 35 mm from A, and triangle AMP has a right angle at M, so MP² = 50² − 35² = 2500 − 1225 = 1275. MP = √1275 ≈ 35.7 mm.
What to study next
Read the comparison of construction steps in two diagrams, then test your explanations with the construction reasoning practice set.
If you want a teacher to ask “why” after each step of your own constructions, see online one-to-one Engineering Drawing tuition.