Question 1208283
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Exact Answer = <font color=red>10*sqrt(129) kilometers</font>
Approximate Answer = <font color=red>113.5782 kilometers</font>
Ask your teacher how s/he wants you to round the approximate value. 


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Explanation



The compass bearing has 000° pointing directly north. 
As you turn eastward, i.e. rotate clockwise, the angle increases.
045° points to the northeast
090° points east
135° points southeast
And so on.
*[illustration UploadedScreenshot_62.png]


With that in mind, here is what the diagram looks like
*[illustration UploadedScreenshot_63.png]
Angle APT = 70° and Angle BTQ = 130° are given
Points A and B are directly north of P and T respectively. 


Angle BTP = 110° is found by solving the equation angleAPT+angleBTP = 180. Note how vertical segments AP and BT are parallel, which means the consecutive interior angles are supplementary.
Angle PTQ = 120° is determined by noting that the three angles around point T must add to 360 (you'll solve this equation: anglePTB+angleBTQ+anglePTQ = 360)


Focus on triangle PTQ.
To find x, the length of segment PQ, we can use the Law of Cosines.
c^2 = a^2 + b^2 - 2*a*b*cos(C)
x^2 = 50^2 + 80^2 - 2*50*80*cos(120)
x^2 = 50^2 + 80^2 - 2*50*80*(-1/2)
x^2 = 12900
x = sqrt(12900)
x = sqrt(100*129)
x = sqrt(100)*sqrt(129)
x = <font color=red>10*sqrt(129) which is the exact distance</font>
x = 113.578166916005
x = <font color=red>113.5782 kilometers which is the approximate distance</font>
Make sure that your calculator is set to degrees mode.
Ask your teacher how s/he wants you to round this approximate value. 


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Another Approach.


Place P at the origin (0,0)
This is how we'll locate point T.
T = P + 50*( sin(70), cos(70) )
T = (0,0) + 50*( 0.9396926, 0.3420201 )
T = ( 50*0.9396926, 50*0.3420201 )
T = ( 46.98463, 17.101005 )
The decimal values are approximate. 
Make sure that your calculator is set to degrees mode.


Normally cosine is associated with the x coordinate; however, the compass bearing angles have 000° pointing north (rather than east), so we have a 90° rotation. This 90° rotation swaps the roles of sine and cosine. Cosine is a 90° phase-shifted version of sine.


We'll follow a similar method to find where point Q is located.
Q = T + 80*( sin(130), cos(130) )
Q = ( 46.98463, 17.101005 ) + 80*( 0.7660444, -0.6427876 )
Q = ( 46.98463, 17.101005 ) + ( 80*0.7660444, 80*(-0.6427876) )
Q = ( 46.98463, 17.101005 ) + ( 61.283552, -51.423008 )
Q = ( 46.98463+61.283552, 17.101005+(-51.423008) )
Q = ( 108.268182, -34.322003 )
The decimal values are approximate. 



Here is the calculation template for point Q's coordinates in one single line
Q = ( 50*sin(70)+80*sin(130), 50*cos(70)+80*cos(130) )
That line is based off of this
Q = P + 50*( sin(70), cos(70) ) + 80*( sin(130), cos(130) )


The key takeaways are these locations
P = (0,0)
Q = ( 108.268182, -34.322003 ) which is approximate


Use the distance formula to find that
PQ = sqrt( (108.268182)^2 + (-34.322003)^2 ) = <font color=red>113.5782 kilometers approximately</font>
The answer will vary depending on the rounding precision.



More practice with similar questions
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and
<a href="https://www.algebra.com/algebra/homework/Triangles/Triangles.faq.question.1182009.html">https://www.algebra.com/algebra/homework/Triangles/Triangles.faq.question.1182009.html</a>
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