Q.):-  In the given figure, ABCD is a rectangle with DC = 8 cm & BC = 6cm.  Line segment CM is drawn making an angle of $30°$ with DC intersecting Ad in M.  Find the length of CM & DM.

 

Dear Student,

Please find below the solution to the asked query:

Given :  ABCD is a rectangle where DC  = 8  cm and BC  = 6 cm 

We know :  $\tan \mathrm{\theta } = \frac{\mathrm{Opposite}}{\mathrm{Adjacent}}$

So, In triangle DCM we get :

$$\begin{gathered} \tan \left(\angle \mathrm{DCM}\right) = \frac{\mathrm{DM}}{\mathrm{DC}} \\ \Rightarrow \tan 30° = \frac{\mathrm{DM}}{8} \\ \mathrm{We} \mathrm{know} :\tan 30° = \frac{1}{\sqrt{3}} , \mathrm{So} \\ \Rightarrow \frac{1}{\sqrt{3}} = \frac{\mathrm{DM}}{8} \\ \Rightarrow \mathrm{DM} = \frac{8}{\sqrt{3}} \\ \Rightarrow \mathrm{DM} = \frac{8}{\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}} \\ \mathbf{\Rightarrow }\mathbf{DM}\mathbf{ }\mathbf{=}\mathbf{ }\frac{\mathbf{8}\sqrt{\mathbf{3}}}{\mathbf{ }\mathbf{3}}\mathbf{ }\mathbf{cm}=\frac{8\times 1.732}{3} =\frac{13.856}{3}=4.6186\mathbf{\approx }\mathbf{4}\mathbf{.}\mathbf{62}\mathbf{ }\mathbf{cm}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{(}\mathbf{ }\mathbf{Ans}\mathbf{ }\mathbf{)} \end{gathered}$$

And we know :  $\mathrm{Cos} \mathrm{\theta } = \frac{\mathrm{Adjacent}}{\mathrm{Hypotenuse}}$

So, In triangle DCM we get :

$$\begin{gathered} \mathrm{Cos} \left(\angle \mathrm{DCM}\right) = \frac{\mathrm{DC}}{\mathrm{CM}} \\ \Rightarrow \mathrm{Cos} 30° = \frac{8}{\mathrm{CM}} \\ \mathrm{We} \mathrm{know} :\mathrm{Cos} 30° = \frac{\sqrt{3}}{2} , \mathrm{So} \\ \Rightarrow \frac{\sqrt{3}}{2} = \frac{8}{\mathrm{CM}} \\ \Rightarrow \mathrm{CM} = \frac{16}{\sqrt{3}} \\ \Rightarrow \mathrm{CM} = \frac{16}{\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}} \\ \mathbf{\Rightarrow }\mathbf{CM}\mathbf{ }\mathbf{=}\mathbf{ }\frac{\mathbf{16}\sqrt{\mathbf{3}}}{\mathbf{ }\mathbf{3}}\mathbf{ }\mathbf{cm}=\frac{16\times 1.732}{3} =\frac{27.712}{3}=9.237\mathbf{\approx }\mathbf{9}\mathbf{.}\mathbf{24}\mathbf{ }\mathbf{cm}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{(}\mathbf{ }\mathbf{Ans}\mathbf{ }\mathbf{)} \end{gathered}$$
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