Question

In the given figure, an altitude is drawn to the hypotenuse .The lengths of different segments are marked in the figure. Determine the values of x, y and z in the given case.

Answer 1

We have to find the values of x, y, z.

$$\begin{gathered} \mathrm{In}△\mathrm{ABC},\mathrm{m}\angle \mathrm{ABC}=90°. \\ \text{S}\mathrm{eg}\mathrm{BD}\perp \text{H}\mathrm{ypotenuse}\mathrm{AC} \\ \therefore {\mathrm{BD}}^2=\mathrm{AD}\times \mathrm{DC}[\mathrm{By}\mathrm{property}\mathrm{of}\mathrm{geometric}\mathrm{mean}] \\ \Rightarrow y^2=4\times 5 \\ \Rightarrow y=\sqrt{4\times 5} \\ \Rightarrow y=2\sqrt{5} \\ \mathrm{In}△\mathrm{ADB}, \\ \mathrm{m}\angle \mathrm{ADB}=90° \\ {\mathrm{AB}}^2={\mathrm{AD}}^2+{\mathrm{BD}}^2[\mathrm{By}\text{P}\mathrm{ythagoras}\text{'}\text{T}\mathrm{heore}\text{m}] \\ {\mathrm{AB}}^2=4^2+y^{\mathit{2}} \\ \Rightarrow \Rightarrow x^2=4^2+(2\sqrt{5}{)}^2 \\ \Rightarrow x^2=16+20 \\ \Rightarrow x^2=36 \\ \Rightarrow x=\sqrt{36} \\ \Rightarrow x=6 \\ \mathrm{In}△\mathrm{BDC}, \\ {\mathrm{BC}}^2={\mathrm{BD}}^2+{\mathrm{CD}}^2[\mathrm{By}\text{Py}\mathrm{thagoras}\text{'}\text{T}\mathrm{he}\mathrm{orem}] \\ \Rightarrow z^2=(2\sqrt{5}{)}^2+5^2 \\ \Rightarrow z^2=20+25 \\ \Rightarrow z^{\mathit{2}}=45 \\ \Rightarrow z=\sqrt{45} \\ \Rightarrow z=3\sqrt{5} \end{gathered}$$