Question

A lamp post is situated at the middle point $M$ of the side $AC$ of a triangular plot $ABC$ with $BC=7m,CA=8cm$ and $AB=9m$. Lamp post subtends angle ${15}^o$ at the point $B$. Determine the height of the lamp post.

Answer 1

R.E.F image

cos c=$\frac{a^2+b^2-c^2}{2ab}$

$=\frac{7^2+8^2-9^2}{2\times 7\times 8}$

$=\frac{49+64-81}{16\times 7}$

= $\frac{32}{16\times 7}$

= $\frac{2}{7}$

In $△$BMC,

cos c = $\frac{C{M}^2+C{B}^2-M{B}^2}{2\times CM\times CB}$

$\frac{2}{7}=\frac{4^2+7^2-M{B}^2}{2\times 4\times 7}$

$\frac{2}{7}=\frac{16+49-M{B}^2}{8\times 7}$ $\therefore$ MB=7

In $△PMB,{90}^{\circ }$ at M

tan 15=$\frac{PM}{MP}\Rightarrow \frac{1-\frac{1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}}=\frac{PM}{7}\Rightarrow PM=(1^4-7\sqrt{3})$