The diagram shows the position of three airports . A and G. G is 200 kilometres from A E is 160 kilometres from A From G the bearing of A is 052 degrees From A the bearing of E is 216 degrees How far apart are airports G and E?
If you have the diagram, you can see that ∡ is 16°
So use the law of cosines.
GE^2 = 160^2 + 200^2 - 2*160*200 cos16°
I got the answer to be 158km but im not sure how to explain it can someone help me
All angles are measured CW from +y-axis.
AG = 200km[52o], GA = 200km[52o+180o].
AE = 160km[216o].
GE = GA + AE = 200[232] + 160[216].
GE = (200*sin232+160*sin216) + (200*cos232+160*cos216)I
-252 - 253i.
GE = sqrt(252^2 + 253^2) =
To find the distance between airports G and E, we can use the concept of bearings and the given information.
Let's start by drawing a diagram:
G
|
|
|
A — — — — — —|— — — — — — E
From the information given, we know that G is 200 kilometers from A, and the bearing of A from G is 052 degrees. We also know that E is 160 kilometers from A, and the bearing of E from A is 216 degrees.
To find the distance between G and E, we can use the concept of trigonometry. Specifically, we can use the Law of Cosines:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, a represents the distance from G to A, b represents the distance from A to E, and C represents the angle between a and b, which is the bearing of A from G (052 degrees) plus the bearing of E from A (216 degrees). However, since bearings are usually measured from north (clockwise), we need to convert the bearing of A from G to its supplementary angle, which is 360 - 52 = 308 degrees.
Now, let's plug in the values into the formula:
c^2 = 200^2 + 160^2 - 2 * 200 * 160 * cos(308 + 216)
To simplify the calculation, we can use a scientific calculator or an online trigonometry calculator to find the cosine of the angle (308 + 216) and solve for c.
c^2 = 40000 + 25600 - 2 * 200 * 160 * cos(524)
c^2 = 65600 - 64000 * cos(524)
c^2 ≈ 65600 - 64000 * (-0.724776)
c^2 ≈ 65600 + 46368.64
c^2 ≈ 111968.64
c ≈ √(111968.64)
c ≈ 334.95
Therefore, the approximate distance between airports G and E is 334.95 kilometers.