Question

Show that the diagonals of a rhombus divide it into four congruent triangles.

Answer 1

$\begin{array}{c}letABCDisarhombuswhoseallsideareequalandAC\perp BD\\ \angle AOB=\angle AOD=\angle COD=\angle COB={90}^0\\ In\Delta ABCand\Delta CBO\\ \Rightarrow AB=BC\\ \Rightarrow AO=CO\\ \Rightarrow \angle BOA=\angle BOC={90}^0\\ bySAScriteria\Delta ABOiscongurentto\Delta CBO\\ SimillaryInrighttriangleAODandCOD\\ \Rightarrow AD=DC\\ \Rightarrow \angle AOD=\angle COD={90}^0\\ \Rightarrow \therefore AO=AD\\ \therefore \Delta AODiscongruentto\Delta COD\\ Hence,allfourtrianglesarecongruent.\end{array}$