Question

Consider a triangular plot ABC with sides AB = 7 m, BC = 5 m and CA = 6 m. A vertical lamp-post at the mid-point D of AC subtends an angle `30^(@)` at B. The height (in m) of the lamp-post is
A. `(2)/(3)sqrt(21)`
B. `2sqrt(21)`
C. `7sqrt(3)`
D. `(3)/(2)sqrt(21)`

Answer 1

Correct Answer - A
According to given information, we have the following figure.

Clearly, length of `BD =(1)/(2)sqrt(2a^(2)+2c^(2)-b^(2))`,
(using Appollonius theorem)
where, c = AB = 7, a = BC = 5
and b = CA = 6
`therefore" "BD =(1)/(2)sqrt(2xx25+2xx49-36)`
`=(1)/(2)sqrt(112)=(1)/(2)4sqrt(7)=2sqrt(7)`
Now, let ED = h be the height of the lamp post.

Then, in `DeltaBDE, tan 30^(@) = (h)/(BD)`
`implies (1)/(sqrt(3))=(h)/(2sqrt(7))`
`implies h = (2sqrt(7))/(sqrt(3))=(2)/(3)sqrt(21)`