Dividing a Line into Equal Parts
≈ 34 minDividing a Line into Equal Parts
Suppose you must divide a 97 mm line into seven equal parts. Dividing by hand gives 13.857… mm — a number you cannot set on a ruler accurately, and seven small errors accumulate along the line.
The construction avoids arithmetic entirely. From one end of AB, draw a line at any convenient acute angle. Step off seven equal divisions of any convenient size along it with the compasses — 10 mm each is easy. Join the last division to B. Then draw lines through the other six divisions parallel to that closing line. Where they cut AB, they divide it into seven exactly equal parts.
Worked example. Whether your stepped-off divisions are 8 mm or 15 mm makes no difference to the result. The parallels transfer the proportion, not the length — which is why the construction works for any awkward division.
Core checkpoint: The stepped divisions must be equal to each other, and every transfer line must be parallel to the closing line. Those two conditions are the whole construction.
In the equal-division construction, what size must the divisions stepped off along the angled line be?
A learner joins the last division to B, then draws the remaining transfer lines at slightly different angles "to make them fit". What goes wrong?
To divide a line into seven equal parts using this construction, how many equal divisions must be stepped off along the angled line?
Why is this construction preferred to simply calculating 97 ÷ 7 and measuring each part along the line?