SOLUTION: The bearing from A to C is S 52°E. the bearing from A to B is N 84°E. the bearing from B to C is S 38° W. A plane flying at 250 mph takes 2.4 hours to go from A to B. Find the d

Question 1200155: The bearing from A to C is S 52°E. the bearing from A to B is N 84°E. the bearing from B to C is S 38° W. A plane flying at 250 mph takes 2.4 hours to go from A to B. Find the distance from A to C
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

A compass bearing of something like S 52°E means we start facing directly south, then we turn 52 degrees toward the east.

Refer to this page for a few examples
http://academic.brooklyn.cuny.edu/geology/leveson/core/linksa/comp.html

This is one way to draw out the diagram.

The diagram is to scale. It was made with GeoGebra.

The distance AB = 600 is due to the scratch work calculation shown below
distance = rate*time
distance = (250 mph)*(2.4 hrs)
distance = 600 miles

Angles EAC, DAB, and GBC are the compass bearings given to us. They are marked in blue shown above.

Angle	Compass Bearing
EAC = 52	S 52°E
DAB = 84	N 84°E
GBC = 38	S 38°W

The last two letters of each angle name tells us where we are located and where we are looking at in that exact order.
Example: Angle EAC has "A" and "C" as the last two letters. We're located at A and look toward C.

Vertical line DE is parallel to vertical line FG.
Segment AB is a transversal cut to these parallel lines.
Same side interior angles DAB and FBA are supplementary (because of the parallel lines).
So,
(angle DAB)+(angle FBA) = 180
(84)+(angle FBA) = 180
angle FBA = 180-84
angle FBA = 96

Angle BAC is found by noticing angles EAC, BAC, and DAB are supplementary

In other words,
(angle EAC)+(angle BAC)+(angle DAB) = 180
and similarly
(angle GBC)+(angle CBA)+(angle FBA) = 180
these two equations help determine angles BAC and CBA (44 and 46 respectively)

Then we use the idea that the inner angles of a triangle add to 180
A+B+C = 180
44+46+C = 180
90+C = 180
C = 90
Triangle ABC is a right triangle.

Since ABC is a right triangle, we have two options using the trig ratios sine and cosine

Option 1	Option 2
sin(angle) = opposite/hypotenuse sin(angle ABC) = AC/AB sin(46) = x/600 x = 600*sin(46) x = 431.60388020319 x = 431.6	cos(angle) = adjacent/hypotenuse cos(angle BAC) = AC/AB cos(44) = x/600 x = 600*cos(44) x = 431.60388020319 x = 431.6

Make sure your calculator is in degree mode.

Here's one possible route if you wanted to use the law of sines
sin(B)/b = sin(C)/c
sin(46)/x = sin(90)/600
600*sin(46) = x*sin(90)
x = 600*sin(46)/sin(90)
x = 431.60388020319
x = 431.6

Keep in mind that sin(90) = 1, so this calculation isn't entirely new compared to option 1 mentioned earlier.

Answer: Approximately 431.6 miles
Round this value however your teacher instructs.

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