249. The Bisecting Diagonals · Construct a parallelogram from side and diagonals

Try to break it

Drag O, B, or C. The other endpoints of the diagonals snap to keep O as the midpoint of both. The resulting quadrilateral is always a parallelogram. Try to drag the handles so O drifts off either midpoint — impossible. Bisecting diagonals always make a parallelogram.

How you build it

Construct a parallelogram from side and diagonals.

• Point tool: mark point A — one end of the first diagonal.
• Point tool: mark point C — the other end of the first diagonal.
• Segment tool: join A to C — the first diagonal.
• Midpoint tool: click A then C — O drops exactly at the centre of diagonal AC.
• Point tool: mark point B — one end of the second diagonal, off the line AC.
• Line tool: click B, then O — extend it past O.
• Arc tool: click O, then B — radius OB.
• Point tool: mark point D where the arc meets the line on the far side from B — now OD = OB, so O bisects both diagonals.
• Segment tool: join A to B.
• Segment tool: join B to C.
• Segment tool: join C to D.
• Segment tool: join D to A. The diagonals bisect each other at O, so ABCD is a parallelogram.

The proof, step by step

Prove that the constructed figure is a parallelogram — its diagonals bisect each other.

• By construction, OB = OD (since D is the extension of BO such that OD = OB).
• By construction, OC = OA (since A is the extension of CO such that OA = OC).
• Therefore, the diagonals AC and BD bisect each other at O.
• A quadrilateral whose diagonals bisect each other is a parallelogram. Hence, ABCD is a parallelogram.

Worked example

If the diagonals of a parallelogram are 10 cm and 12 cm, what is the length of the segment from the intersection to the vertex on the 10 cm diagonal?

The diagonals of a parallelogram bisect each other. So the segment from the intersection to the vertex on the 10 cm diagonal is half of 10 cm, which is 5 cm.

• 5 cm — correct
• 6 cm
• 10 cm
• 12 cm