169. The Rhombus Cross · diagonals that always bisect at right angles

Try to break it

Drag A along the horizontal diagonal and B along the vertical diagonal. The diagonals always cut each other in half AND meet at a perfect 90°. Try to find a setting where O isn't the midpoint of both, or where the diagonals aren't perpendicular — impossible.

How you build it

Construct a rhombus from perpendicular diagonals.

• Mark O — the point where both diagonals will cross at 90°. Both diagonals will be bisected here.
• Mark vertex A. The distance OA becomes one half of diagonal AC.
• Draw diagonal line 1 through O and A. Click O, then click A. This line will give vertex C on the far side.
• Use the Perp tool to draw diagonal line 2 through O at exactly 90° to line OA. Click on O (on the diagonal), then click to one side.
• With centre O, draw an arc of radius OA. Click O first, then click A. The arc will cross diagonal line 1 at A and at C — directly opposite A through O.
• Mark C where the arc meets diagonal line 1 on the far side from A. Click that crossing — the canvas snaps to the exact point where OC = OA.
• With centre O, draw a new arc — this sets the length OB. Click O first, then click anywhere on the perpendicular line to choose your radius. The arc crosses the perpendicular at B and D.
• Mark B where the new arc meets the perpendicular line. Click one of the crossings — the canvas snaps to the exact intersection.
• Mark D where the same arc meets the perpendicular on the OTHER side from B. Now OD = OB, so both diagonals bisect at O and meet at 90°. ABCD must be a rhombus.
• 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 — connect D back to A. Since OA=OC, OB=OD, and AC⊥BD, all four sides are equal. That is a rhombus!

The proof, step by step

Prove that the diagonals of a rhombus bisect each other at right angles.

• In ΔAOB and ΔCOD: AB = CD (rhombus sides), ∠OAB = ∠OCD (alt. angles), ∠OBA = ∠ODC (alt. angles). ∴ ΔAOB ≅ ΔCOD (ASA) ⇒ OA = OC, OB = OD.
• In ΔAOB and ΔCOB: OA = OC (proved), OB = OB (common), AB = BC (rhombus sides). ∴ ΔAOB ≅ ΔCOB (SSS) ⇒ ∠AOB = ∠COB.
• ∠AOB + ∠COB = 180° (linear pair on AC). Since ∠AOB = ∠COB, each must be 90°. ∴ Diagonals intersect at right angles.

Worked example

In a rhombus ABCD, diagonals AC and BD intersect at O. If AC = 24 cm and BD = 10 cm, what is the length of each side of the rhombus?

Diagonals of a rhombus bisect each other at 90°. So OA = 12 cm, OB = 5 cm. In right ΔAOB, AB² = OA² + OB² = 144 + 25 = 169. Thus AB = 13 cm.

• 12 cm
• 13 cm — correct
• 14 cm
• 15 cm