Problem
GEO-B2-M09-P017 A Rhombus from Coordinates
#17
★★★☆☆ Level 3 of 5
Let \(A(-a,0)\), \(B(0,b)\), \(C(a,0)\), \(D(0,-b)\), where \(a,b>0\). Prove that \(ABCD\) is a rhombus and its diagonals are perpendicular.
Compare squared side lengths. The diagonals lie on the coordinate axes.
\(AB^2=BC^2=CD^2=DA^2=a^2+b^2\), so all sides are equal and \(ABCD\) is a rhombus. Diagonal \(AC\) lies on the \(Ox\)-axis, while \(BD\) lies on the \(Oy\)-axis, hence \(AC\perp BD\).
Here coordinates prove both equality of sides and perpendicularity.