Question

The sides of a right-angled triangle containing the right angle are 9 and 12 cm respectively. Find the length of the perpendicular from the right angle of the hypotenuse.

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{ABC}\mathrm{be}\mathrm{the}\mathrm{right}∆\mathrm{in}\mathrm{which}\angle \mathrm{B}=90°. \\ \mathrm{Now},\mathrm{AB}=9\mathrm{cm};\mathrm{BC}=12\mathrm{cm} \\ \mathrm{now},\mathrm{AC}=\sqrt{{\mathrm{AB}}^2+{\mathrm{BC}}^2}=\sqrt{{\left(9\right)}^2+{\left(12\right)}^2}=15\mathrm{cm} \\ \mathrm{area}\mathrm{of}∆\mathrm{ABC}=\frac{1}{2}\times \mathrm{BC}\times \mathrm{AB}=\frac{1}{2}\times 12\times 9=54{\mathrm{cm}}^2 \\ \mathrm{Let}\mathrm{BD}\perp \mathrm{AC}. \\ \mathrm{Now},\mathrm{area}\mathrm{of}∆\mathrm{ABC}=54{\mathrm{cm}}^2 \\ \Rightarrow \frac{1}{2}\times \mathrm{AC}\times \mathrm{BD}=54 \\ \Rightarrow \frac{1}{2}\times 15\times \mathrm{BD}=54 \\ \Rightarrow \mathrm{BD}=\frac{54\times 2}{15} \\ \Rightarrow \mathrm{BD}=7.2\mathrm{cm} \end{gathered}$$