question_answer

The area of a rhombus whose perimeter is 80 m and one of whose diagonal is 24 m, is

A)  380 \[{{m}^{2}}\]                              

B)  370\[{{m}^{2}}\]

C)  374 \[{{m}^{2}}\]                              

D)  384\[{{m}^{2}}\]

Correct Answer: D

Solution :

(d): Let ABCD be the rhombus of perimeter 80 m and diagonal \[AC=24\]m We have, \[AB+BC+CD+DA=80\] \[\Rightarrow \]\[4AB=80\]         \[\left[ \therefore A.B=BC=CD=DA\text{ } \right]\] \[\Rightarrow \]\[AB=20\]m In \[\Delta ABC\], we have \[2s=AB+BC+AC=20+20++24=64\] \[\Rightarrow s=32\] \[\therefore \]\[{{\Delta }_{1}}=Area\,of\,\Delta ABC=\sqrt{32\times 12\times 12\times 8}\]\[=16\times 12=192\,{{m}^{2}}\] Hence, area of rhombus \[ABCD=2\times 192\,{{m}^{2}}=384\,{{m}^{2}}\].