Trigonometrical Ratios And Identities Ques 8
- Consider a triangular plot $A B C$ with sides $A B=7 \mathrm{m}$, $B C=5 \mathrm{m}$ and $C A=6 \mathrm{m}$. A vertical lamp-post at the mid-point $D$ of $A C$ subtends an angle $30^{\circ}$ at $B$. The height (in $\mathrm{m}$ ) of the lamp-post is
(2019 Main, 10 Jan I)
(a) $\frac{2}{3} \sqrt{21}$
(b) $2 \sqrt{21}$
(c) $7 \sqrt{3}$
(d) $\frac{3}{2} \sqrt{21}$
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Answer:
Correct Answer: 8.(a)
Solution: (a) According to given information, we have the following figure.
Clearly, length of $B D=\frac{1}{2} \sqrt{2 a^2+2 c^2-b^2}$,
(using Appollonius theorem)
where, $c=A B=7, a=B C=5$
and $b=C A=6 $
$B D=\frac{1}{2} \sqrt{2 \times 25+2 \times 49-36} $
$=\quad \frac{1}{2} \sqrt{112}=\frac{1}{2} 4 \sqrt{7}=2 \sqrt{7}$
Now, let $E D=h$ be the height of the lamp post.
Then, in $\triangle B D E, \tan 30^{\circ}=\frac{h}{B D}$
$\Rightarrow \quad \frac{1}{\sqrt{3}}=\frac{h}{2 \sqrt{7}}$
$\Rightarrow \quad h=\frac{2 \sqrt{7}}{\sqrt{3}}=\frac{2}{3} \sqrt{21}$
BETA
