Two houses stand just on the opposite sides of a road of a village. A ladder that stands against the wall of the first house is at a distance of 6 metres from the house and makes an angle of `30^(@)` with the horizontal line.But if the ladder stands against the wall of the second house keeping its foot at the same point, it makes an angle of `60^(@)` with the horizontal line.
(a) Find the length of the ladder.
(b) Calculate the distance of the foot of the ladder from the foot of the wall of the second house.
(c) Find the width of the road.
Find the height where the top of the ladder is fixed against the wall of the second house.

Explanation

Let AB and CD be two houses so that BD is the width of the road. The ladder PQ makes an angle of `30^(@)`when it is against the first house AB and the same ladder PR madkes an angle of `60^(@)` when it is against the second house CD.
`therefore angleBPQ = 30^(@) and angleDPR = 60^(@)`
As per question, BP = 6 metres
Now, from the right-angled triangle BPQ we get,
`cos 30^(@) = (BP)/(PQ)` [ by definition ]
or, `(sqrt(3))/(2) = (6)/(PQ) " " or, sqrt(3)PQ = 12 " " or, " " PQ = (12)/(sqrt(3)) = (4sqrt(3).sqrt(3))/(sqrt(3)) = 4sqrt(3)`
Hence the length of the ladder =`4sqrt(3)` metres.
Again. from the right-angled triangle DPR we get,
`cos60^(@) =(PD)/(PR)` [ by definition ]
or, `(1)/(2) =(PD)/(PQ) [because` PR = PQ ]
or,` " " (1)/(2) = (PD)/(4sqrt(3)) " "or, " " 2PD = 4sqrt(3) " " or, PD =(4sqrt(3))/(2) or, PD = 2sqrt(3)`.
`therefore " " BD = BP + PD = (6 + 2sqrt(3))`m.
Also, from the right-angled triangle PDR we get,
`sin 60^(@) = (DR)/(PR)` [ by definition ]
or, `(sqrt(3))/(2) = (DR)/(PQ) [ because PR = PQ ]`
or, `(sqrt(3))/(2) = (DR)/(4sqrt(3)) " "or, 2DR = 12 " "or, DR = (12)/(2) = 6`
Hence the ladder is fixed against the wall of the second houe at a height of 6 metres above the ground.
Hence (a) the length of the ladder is `4sqrt(3)` m.
(b) the distance of the foot of the ladder from the foot of the wall of the second house is `2sqrt(3)`m.
(c) the width of the road is `2(3 + sqrt(3))` m.
(d) the height where the top of the ladder is fixed against the wall of the second house is 6 m.