Question

ABCD is a rhombus whose diagonals intersect at O. If AB = 10 cm, diagonal BD = 16 cm, find the length of diagonal AC.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{know}\mathrm{that}\mathrm{the}\mathrm{diagonals}\mathrm{of}\mathrm{a}\mathrm{rhombus}\mathrm{bisect}\mathrm{each}\mathrm{other}\mathrm{at}\mathrm{right}\mathrm{angles}. \\ \therefore \mathrm{BO}=\frac{1}{2}\mathrm{BD}=(\frac{1}{2}\times 16)\mathrm{cm} \\ =8\mathrm{cm} \\ \mathrm{AB}=10\mathrm{cm}\mathrm{and}\angle \mathrm{AOB}=90° \\ \mathrm{From}\mathrm{right}∆\mathrm{OAB}: \\ {\mathrm{AB}}^2={\mathrm{AO}}^2+{\mathrm{BO}}^2 \\ \Rightarrow {\mathrm{AO}}^2=({\mathrm{AB}}^2–{\mathrm{BO}}^2) \\ \Rightarrow {\mathrm{AO}}^2=(10{)}^2-(8{)}^2{\mathrm{cm}}^2 \\ \Rightarrow {\mathrm{AO}}^2=(100-64){\mathrm{cm}}^2=36{\mathrm{cm}}^2 \\ \Rightarrow \mathrm{AO}=\sqrt{36}\mathrm{cm}=6\mathrm{cm} \\ \therefore \mathrm{AC}=2\times \mathrm{AO}=(2\times 6)\mathrm{cm}=12\mathrm{cm} \end{gathered}$$