The side of the rhombus is 9, and the distance from the center of the rhombus to it is 1. Find the area of the rhombus

Denote the key points as shown.
We continue the height of OE to side AB and denote the intersection point by F.
The area of the rhombus (as well as the parallelogram) is equal to the product of the height on the side of the rhombus.
Rhombus height = EF (because EF is perpendicular to CD). Consider the triangles DOE and BOF.
DO = OB (by the second property of the rhombus)
∠DOE = ∠BOF (as they are vertical)
∠EDO = ∠FBO (as these are internal cross-lying)
Therefore, the triangles DOE and BOF are equal in the second feature.
Then OE = OF => EF = 2 * OE = 2 * 1 = 2
S diamond = EF * CD = 2 * 9 = 18
Answer: S rhombus = 18

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