Question 1120836: A bridge is going to be built across a canyon from point A to point B. The bearing from A to B is S 28o W. Point C is 87 miles from point A. From point C, the bearing to point A is S 84o E and to point B is S 32o E. Find the required length of the bridge.
Round your answer to two decmial places
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! this is a bit of a messy problem to solve, but i think i have it.
i tried to make the diagram as simple to follow as possible.
it is shown below:
from A to be, the bearing is south 28 degrees west.
that would be angle BAD.
C is 87 miles from A.
that would be line segment AC.
from C, the bearing to A would be south 84 degrees east.
that would be angle ACF.
from C, the bearing to B would be south 32 degrees east.
that would be angle BCE.
triangle CFA is a right triangle with the 90 degree angle being AFC.
since angle ACF is 84 degrees, that makes angle CAF equal to 6 degrees because the sum of the acute angles of a right triangle = 90 degrees and 84 + 6 = 90.
angle FAD is a 90 degree angle because it is the intersection of horizontal line FA and vertical line DA.
angle CAD is therfore equal to to 68 degrees because 62 + 6 = 68.
angle BCA is equal to 52 degrees because it is the difference of angle ECA and angle ECB. in other words, 84 - 32 = 52.
you have triangle CAB formed with angle BCA = 52 degrees and angle BAC = 68 degrees.
since the sum of the angles of a triangle = 180, this makes angle CBA equal to 60 degrees because 180 - 52 - 68 = 60.
your triangle CAB contains angles angle BCA = 52 degrees, angle CAB = 68 degrees and angle CBA = 60 degrees plus side CA = 87 miles.
from here, you can use the law of sines to find the distance between point B and A which is side BA of triangle CAB.
the law of sines states a/sin(A) = b/sin(B) = c/sin(C).
using this law, you get 87 / sin(60) = BA / sin(52).
solve for BA to get BA = 87 * sin(52) / sin(60).
this results in BA = 79.16273041 miles.
i did lots of double checking to see if i got it right and i confirmed that this is the solution, so i'm reasonably confident that it is the number you are looking for.
round the answer to decimal places and it should be 79.16 miles.
here's a referencee you might find useful.
it tells you what bearing means, but you have to use some geometric and /or trigonometric logic to get the angles you need and you need to be able to use the law of sines or the law of cosines to get your final answer.
https://www.mathsteacher.com.au/year7/ch08_angles/07_bear/bearing.htm
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