Question

Mark (✓) against the correct answer:
Each side of a rhombus is 15 cm and the length of one of its diagonals is 24 cm. The area of the rhombus is
(a) 432 cm²
(b) 216 cm²
(c) 180 cm²
(d) 144 cm²

Answer 1

(b) 216 cm²


$$\begin{gathered} \mathrm{Let}ABCD\mathrm{be}\mathrm{a}\mathrm{rhombus}\mathrm{whose}\mathrm{diagonals}AC\mathrm{and}BD\mathrm{intersect}\mathrm{at}\mathrm{a}\mathrm{point}O. \\ \mathrm{Let}\mathrm{the}\mathrm{length}\mathrm{of}\mathrm{the}\mathrm{diagonal}AC\mathrm{be}24\mathrm{cm}\mathrm{and}\mathrm{the}\mathrm{side}\mathrm{of}\mathrm{the}\mathrm{rhombus}\mathrm{be}15\mathrm{cm}. \\ \mathrm{We}\mathrm{know}\mathrm{that}\mathrm{the}\mathrm{diagonals}\mathrm{of}\mathrm{the}\mathrm{rhombus}\mathrm{bisect}\mathrm{each}\mathrm{other}\mathrm{at}\mathrm{right}\mathrm{angles}. \\ \therefore AO=\frac{1}{2}AC \\ \Rightarrow AO=\left(\frac{1}{2}\times 24\right)\mathrm{cm} \\ \Rightarrow AO=12\mathrm{cm} \\ \mathrm{From}\mathrm{right}∆AOB,\mathrm{we}\mathrm{have}: \\ B{O}^2=A{B}^2-A{O}^2 \\ \Rightarrow B{O}^2=\left\{{\left(15\right)}^2-{\left(12\right)}^2\right\} \\ \Rightarrow B{O}^2=\left(225-144\right) \\ \Rightarrow B{O}^2=81 \\ \Rightarrow BO=\sqrt{81} \\ \Rightarrow BO=9\mathrm{cm} \\ \therefore BD=2\times BO \\ BD=\left(2\times 9\right)\mathrm{cm} \\ BD=18\mathrm{cm} \\ \mathrm{Hence},\mathrm{the}\mathrm{length}\mathrm{of}\mathrm{the}\mathrm{other}\mathrm{diagonal}\mathrm{is}18\mathrm{cm}. \\ \mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{rhombus}=\left(\frac{1}{2}\times 24\times 18\right){\mathrm{cm}}^2 \end{gathered}$$

$=216{\mathrm{cm}}^2$