Special Types of Quadrilaterals

167. The Diagonals' Pact · always meeting at their midpoints

OA always equals OC, and OB always equals OD.

In a parallelogram, the diagonals bisect each other — they cross at point O which is the midpoint of both diagonals. So OA = OC and OB = OD. This is one of the four key properties of a parallelogram.

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Selina ICSE: Special Types of Quadrilaterals

What this lesson covers

Try to break it

Drag A, B, or C. The fourth vertex D follows to keep ABCD a parallelogram, and the diagonals AC and BD always cross at their exact midpoint. Try to drag a vertex so the diagonals miss the midpoint — impossible.

How you build it

Construct a parallelogram from bisecting diagonals.

Mark O — the point where both diagonals will cross. Both diagonals will be bisected at O, so OA = OC and OB = OD.
Mark vertex A anywhere. OA will become one half of diagonal AC. The other half OC will be equal to OA.
Draw a segment through O and A — this is part of the first diagonal. The compass arc will find C on the other side.
With centre O, draw a full arc of radius OA. Click O first, then click A. The arc will cross the diagonal line at A and at C — exactly opposite A through O.
Mark C where the arc meets the diagonal on the far side from A. Click that crossing — OC = OA.
Mark B anywhere — but NOT on the line OA. B starts the second diagonal.
Draw a segment through O and B — the second diagonal. The compass arc will find D on the other side.
With centre O, draw an arc of radius OB. Click O first, then click B. The arc crosses the second diagonal at B and at D.
Mark D where the arc meets the second diagonal on the far side from B. Now OD = OB.
Draw side AB — connect vertex A to vertex B.
Draw side BC — connect vertex B to vertex C.
Draw side CD — connect vertex C to vertex D.
Draw side DA — ABCD is a parallelogram because OA = OC and OB = OD!

The proof, step by step

Prove that the diagonals of the parallelogram bisect each other.

In ΔAOB and ΔCOD, ∠OAB = ∠OCD (Alternate angles, AB || CD)
∠OBA = ∠ODC (Alternate angles, AB || CD)
AB = CD (Opposite sides of a parallelogram are equal)
∴ ΔAOB ≅ ΔCOD (ASA Congruence)
∴ OA = OC and OB = OD (Corresponding parts of congruent triangles are congruent)

Worked example

In a parallelogram ABCD, the diagonals AC and BD intersect at O. If AC = 24 cm and BD = 18 cm, what is the length of OA?

The diagonals of a parallelogram bisect each other. Therefore, OA is exactly half of AC. OA = 24 / 2 = 12 cm.

6 cm
8 cm
12 cm — correct
18 cm