question_answer 20)

The diagonals of a rhombus measure 16 cm and 30 cm. Find its perimeter.      

Answer:

                Let ABCD be a rhombus whose diagonals BD and AC are of lengths 16 cm and 30 cm respectively. Let the diagonals BD and AC intersect each other at O.                            Since the diagonals of a rhombus bisect each other at right angles. Therefore                                            \[\text{BO}=\text{OD}=\text{8 cm},\] \[\text{AO}=\text{OC}=\text{15 cm},\] \[\angle AOB=\angle BOC=\angle COD=\angle DOA={{90}^{o}}\]In right-angled triangle AOB. \[A{{B}^{2}}=O{{A}^{2}}+O{{B}^{2}}\] \[\left| \text{By Pythagoras Property} \right.\]  \[\Rightarrow \]        \[\text{A}{{\text{B}}^{\text{2}}}=O{{A}^{\text{2}}}+O{{B}^{\text{2}}}\]  \[\Rightarrow \]              \[~\text{A}{{\text{B}}^{\text{2}}}={{15}^{2}}\text{+ }{{\text{8}}^{2}}\] \[\Rightarrow \]               \[A{{B}^{2}}=225+64\] \[\Rightarrow \]               \[A{{B}^{2}}=289\] \[\Rightarrow \]               \[AB=\text{17 cm}\] Therefore, perimeter of the rhombus ABCD \[=\text{4}\]side \[=\text{4}\] AB \[=\text{4}\times \text{17 cm}\] \[=\text{68 cm}\] Hence, the perimeter of the rhombus is 68 cm.