243. The Rhombus Cross · diagonals always meet at 90°

Try to break it

Drag A along the horizontal diagonal to stretch d₁, and B along the vertical diagonal to stretch d₂. The two diagonals always meet at right angles at O, no matter how unequal the lengths get. Try to find a setting where the angle at O drifts off 90° — impossible.

How you build it

Construct a rhombus from perpendicular diagonals.

• Point tool: mark point A — one end of a diagonal.
• Point tool: mark point C — the other end of that diagonal.
• Segment tool: join A to C — the first diagonal.
• Midpoint tool: click A then C — O is where the diagonals will cross.
• Bisector tool: click A then C — the perpendicular bisector of AC. It runs through O, square to AC, and extends both ways; the second diagonal lies on it.
• Arc tool: centre O, drag out — the arc crosses the perpendicular at two points, equal distances from O.
• Point tool: mark B where the arc meets the perpendicular on one side.
• Point tool: mark D where the arc meets the perpendicular on the other side. OB = OD, so BD is bisected too.
• Segment tool: join A to B.
• Segment tool: join B to C.
• Segment tool: join C to D.
• Segment tool: join D to A — all four sides are equal, so ABCD is a rhombus.

The proof, step by step

Prove that the diagonals of a rhombus intersect at right angles.

• ABCD is a rhombus, so all sides are equal: AB = BC.
• Diagonals of a parallelogram bisect each other, so OA = OC.
• In ΔAOB and ΔCOB: AB = BC, OA = OC, OB = OB (common). By SSS, ΔAOB ≅ ΔCOB.
• By CPCTC, ∠AOB = ∠BOC. Since they form a linear pair (180°), each must be 90°.

Worked example

In a rhombus ABCD, the diagonals AC and BD intersect at O. If AC = 10 cm and BD = 24 cm, what is the measure of ∠AOB?

The diagonals of a rhombus always bisect each other at right angles. Therefore, ∠AOB = 90° regardless of the diagonal lengths.

• 45°
• 60°
• 90° — correct
• 120°