Question

A lamp post is situated at the middle point m of the side ac of a triangular plot ABC with BC=7CM,CA=8M AND AB=9M. Lamp posts subtends an $\angle {15}^{\circ }$ at the point B. Determine the height of the lamppost.

Answer 1

In figure,
AB = 9 = c,
BC = 7 = a
AC = 8 = b
Applying cosinc formula in $\Delta ABC$,
$cosC=\frac{a^2+b^2-c^2}{2ab}=\frac{49+64-81}{2\times 7\times 8}=\frac{2}{7}$ . . . . .(i)
Similarly, using cosine formula in $\Delta BMC$,
$B{M}^2=B{C}^2+C{M}^2-2BC.CMcosC$
Here,
$CM=\frac{CA}{2}=4\cdots \{Misthemid-pointofAC\}$
Using, (i), we have
$B{M}^2=49+16-2\times 7\times 4\times \frac{2}{7}=49$
$\Rightarrow BM=7$
Now, In $\Delta BMP$, we have
$tan\theta =\frac{PM}{BM}=\frac{h}{7}$
$\Rightarrow \frac{h}{7}=tan15=2-\sqrt{3}$
$\Rightarrow h=7(2-\sqrt{3})$