Prove that the diagonals of a rhombus are perpendicular

by RoRi

April 28, 2016

Prove that the diagonals of a rhombus are perpendicular.

Proof. For any rhombus, we can take the vertices to be the origin, and the points $A,B$ , and $A + B$ in $\mathbb{R}^2$ . So, the diagonals are the vectors $A+B$ and $A-B$ . Then,

$\begin{align*} (A+B) \cdot (A-B) &= A \cdot A - B \cdot B \\ &= \lVert A \rVert^2 - \lVert B \rVert^2 \\ &= 0 \end{align*}$

since the edges of a rhombus are of equal length $. \qquad \blacksquare$

Stumbling Robot

Prove that the diagonals of a rhombus are perpendicular

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