Trig

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a sailor out in a lake sees two likght houses 11km apart along the shore and gets bearings of 285degrees from his present position for light house A and 237degrees for light house B. From light house B, light house A has a bearing of 45degrees. How far to the nearest kilometre, is the sailor from each light house? What is the shortest distance, the nearest kilometre, from the sailor to the shore? ( the answers are 3km, 13km, 3km)

• Trig -

The hardest part is drawing the diagram.
I have AB at 45° and length 11
point S (for sailor) is to the right, slightly below A so that angle ASB = 48°.
angle B = 12° and angle A = 120°

( I got this by drawing NS-EW lines at A, at B and at S, then using your bearings and properties of parallel lines)

so we have a/sin12 = 11/sin48
a = 3.08
and
b/sin120 = 11/sin48
b = 12.82

for the shortest distance to the shore we have to assume the the shore continues along the line of AB.
We have a nice right-angled triangle where
sin 60 = x/3.08
x = 2.67