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Question 1195837: A and B are two lorry stations 600m apart. A fire station C in on the bearing 042° from the western station at A and 023° from the eastern station at B. (i) Sketch a triangular diagram to illustrate the information above. (ii) Calculate how far is the fire station from the western station at A and from the eastern station at B.
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Part (i)
This is one potential diagram
Points A,B,C were defined earlier in the instructions.
D and E are directly north of A and B respectively.
These two additional points help set up the bearing angles:
angle DAC = 42 degrees (red)
angle EBC = 23 degrees (blue)
A bearing angle of 042° means that we start facing directly north and turn 42 degrees eastward (i.e. clockwise). A similar situation happens with 023°
If we wanted to aim west, then we'd have to use bearings between 180 and 360 degrees.
Eg: A bearing of 270° tells to aim directly west
Another example: the bearing 225° points directly southwest.
One more example: to aim directly northwest, the bearing would be 315°
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Part (ii)
An assumption is that B is directly east of point A.
If that is true, then segment AB is horizontal.
Furthermore, it means angle DAB is 90° which leads to angle CAB being 90-42 = 48 degrees.
With AB being horizontal, we can also say that angle ABC is...
ABC = ABE + EBC
ABC = 90 + 23
ABC = 113 degrees
Here is what things look like with those angle measures added and we erase points D and E since they aren't relevant anymore.
The dashed lines are erased as well.
I've also added in the measure of angle C since C = 180-A-B = 180-48-113 = 19
This sketch is optional, but I find it helps for visual learners.
A,B,C stand for the point locations and they also represent the angle measurements when viewed with a slightly different context. The lowercase counterparts a,b,c will be the sides opposite said angles.
side 'a' is opposite angle A
side 'b' is opposite angle B
side 'c' is opposite angle C
We already know that side c = 600 meters
Let's find side b using the law of sines
sin(B)/b = sin(C)/c
sin(113)/b = sin(19)/600
600*sin(113) = b*sin(19)
b = 600*sin(113)/sin(19)
b = 1696.42793531929
This decimal value is approximate and be sure to follow all instructions on how to round it.
This is the approximate distance from A to C.
Use similar logic to find side 'a'
sin(A)/a = sin(C)/c
sin(48)/a = sin(19)/600
600*sin(48) = a*sin(19)
a = 600*sin(48)/sin(19)
a = 1369.56544791641
This value is approximate as well, and it's the distance from B to C.
This is what the fully solved triangle looks like
The phrase "solve the triangle" means "find all three side lengths and all three angle measures".
The app GeoGebra is a free tool in which you can use the verify the answers.
I use the tool all the time, and it's what I used to make the diagrams above.
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