Trigonometry-12

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A fire at a building B is reported to two fire stations P and Q, 12 km apart, on a straight road.

P observes that the fire is at an angle of 60⁰ to the road while Q observes it to be at an angle of 45⁰.

Which station would have to send the team and how much will it have to travel?

Solution: Let R be the foot of the perpendicular from B on PQ

BR = BP sin 60⁰ = BP x √3 / 2

BR = BQ sin 45⁰ = BQ x 1/√2

Therefore, BP < BQ (since sin 60⁰ > sin 45⁰)

Therefore, the station P will have to send its team.

QR = BR x cot 45⁰

= BR

= BP sin 60⁰

PR = BP cos 60⁰

PQ = PR + QR = BP (cos 60⁰ + sin 60⁰)

= BP [1/2 + √3/2] = BP (√3 + 1) / 2

BP (√3 + 1) / 2 = 12

Hence BP = 2 x 12 / (√3 + 1) = 24 / 2.732 = 8.82 km