246. The Four-Sided Puzzle · Construct any quadrilateral from 4 sides & 1 angle

Try to break it

Drag B to move the whole figure, and C around B's circle to rotate side BC. The four side lengths and the angle at B stay fixed, so the quadrilateral is rigid (up to position and orientation). Three sides plus two angles — or four sides plus one diagonal — pin a quadrilateral uniquely.

How you build it

Construct quadrilateral ABCD with BC = 6 cm, ∠B = 60°, AB = 5 cm, CD = 5.5 cm and DA = 4.5 cm.

• Point tool: mark the vertex B.
• Ray tool: click B, then click to the right — a base ray from B that BC will lie along, and that ∠B is measured from.
• Angle tool: click B, click along the base ray, then set ∠B = 60° with the −/+ buttons — the second arm rises toward A.
• Arc tool: open the compass on the ruler — click the 0 mark, then the 6 mark — and the arc is drawn centred on B, cutting the base ray at C (BC = 6 cm).
• Point tool: mark C where the arc cuts the base ray.
• Arc tool: open the compass on the ruler — click the 0 mark, then the 5 mark — and the arc is drawn centred on B, cutting the angle arm at A (BA = 5 cm).
• Point tool: mark A where the arc cuts the angle arm.
• Arc tool: open the compass on the ruler — click the 0 mark, then the 4.5 mark — and swing the arc centred on A (this is side DA).
• Arc tool: open the compass on the ruler — click the 0 mark, then the 5.5 mark — and swing the arc centred on C; it cuts the previous arc at D.
• Point tool: mark D where the two arcs cross — its position is forced, so the shape is unique.
• Segment tool: join A to D.
• Segment tool: join C to D — quadrilateral ABCD is complete.

The proof, step by step

Prove that the constructed quadrilateral has the given side lengths and angle.

• We start by fixing side BC. This gives us a stable base and two fixed points, B and C.
• At B, we construct the given angle (45°). The ray defines the direction of side AB.
• Using the compass, we cut off length AB on the ray. Point A is now fixed at the correct distance from B.
• Point D must be exactly DA away from A and DC away from C. Two circles (arcs) with these radii intersect at exactly one valid point for the quadrilateral.
• Joining AD and CD completes the figure. The construction is unique because four sides and one included angle rigidly define a quadrilateral.

Worked example

In the construction of quadrilateral ABCD with AB=3.5cm, BC=4.0cm, CD=5.0cm, DA=4.0cm and ∠B=45°, why is it necessary to draw arcs from BOTH A and C to locate D?

One arc from A (radius DA) intersects the plane in a circle, giving infinite possible locations for D. The second arc from C (radius CD) intersects that circle at exactly two points (reflections). In a standard convex quadrilateral construction, the intersection that maintains the correct orientation and doesn't self-intersect is chosen. Thus, both arcs are needed to uniquely fix D.

• Because a single arc from A would give two possible points for D, and the arc from C selects the correct one. — correct
• Because the compass tool only works when drawing two arcs simultaneously.
• Because BC must be parallel to AD, which requires two arcs to verify.
• Because the angle at B must be bisected by the arcs.