Geometry
Why constructions work
Explain what each compass step guarantees, and decide when a triangle can be constructed at all.
A construction is not a drawing done neatly. Every step of it is a claim, and the claim is what makes the finished figure exact rather than approximate. Nothing here asks you to pick up a pair of compasses — the question is always the same one: after this step, what do you now know for certain?
One opening, one guarantee
A compass records a distance. Swing an arc from A and every point on it is exactly that far from A. Swing the same arc from B and every point on it is that far from B. So where the two arcs cross is a point the same distance from A and from B — and that, not neatness, is why the line through the crossings hits the mid-point at a right angle.
Worked example
Why does the angle bisector construction really cut the angle in half?
Equal arcs from the vertex mark points P and Q, one on each arm, with the vertex equally far from both.
Same compass opening, so the two marked lengths are equal — that is the first pair of equal sides.
Try it together
Two sides of a triangle are 5 and 8. Which whole-number lengths could the third side be?
Two conditions fence it in from above and from below.
1.The third side must be less than the other two added together. What is 5 + 8?
Three angles are not enough
Give three angles and you fix the triangle's shape but nothing about its size — endlessly many triangles fit. To pin one down you need at least one length: three sides, or two sides and the angle between them, or two angles and the side between them.
Have a go
Have a go on your own: which of these sets of three lengths can be built into a triangle?
Ready to practise?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.