Three points X,Y and Z lying in the same horizontal plane are such that the bearing of X and Y from Z are 300 and 280 respectively,/YZ/=13m if the bearing of Y from X is 220 Degrees, find the bearing of Z from Y.
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If the bearing of Y from Z is 280, then
the bearing of Z from Y is 280-180 = 100
To find the bearing of Z from Y, we need to use the property that the sum of the bearings should be 360 degrees.
Let's set up the equation:
Bearing of X from Z + Bearing of Y from Z + Bearing of Z from Y = 360 degrees
We know that the bearing of X from Z is 300 degrees and the bearing of Y from Z is 280 degrees. Let's plug in these values:
300 + 280 + Bearing of Z from Y = 360
Combine like terms:
580 + Bearing of Z from Y = 360
Subtract 580 from both sides of the equation:
Bearing of Z from Y = 360 - 580
Simplify:
Bearing of Z from Y = -220 degrees
Therefore, the bearing of Z from Y is -220 degrees. Note that this negative bearing indicates a direction opposite to the conventional orientation.
To find the bearing of Z from Y, we can use the concept of bearings and the given information about the three points X, Y, and Z.
1. Recall that bearing is the angle measured clockwise from the north line to the line joining two points.
2. Let's start by visualizing the situation. Draw a diagram with three points X, Y, and Z in a horizontal plane. Label the given bearings and the distance YZ.
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N
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Z
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Y------X
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3. Given that the bearing of X from Y is 220 degrees, draw a line segment from Y to X with an angle of 220 degrees clockwise from the north line. Now, we have formed a triangle YXZ.
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Z
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Y
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4. We know that the bearing of X from Z is 300 degrees. Since bearings are clockwise angles from the north line, we need to subtract 300 degrees from the bearing of X from Y (220 degrees) to get the internal angle at Y (angle XZY).
Bearing of X from Y = 220 degrees
Bearing of X from Z = 300 degrees
Therefore, angle XZY = (Bearing of X from Y) - (Bearing of X from Z)
= 220 degrees - 300 degrees
= -80 degrees
Note: Since the bearings are measured clockwise and anti-clockwise from the north line, we have a negative angle here.
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N
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Z
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| \ Y
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5. We now have a triangle XYZ. We can use the Law of Cosines to find the bearing of Z from Y.
The Law of Cosines states: c^2 = a^2 + b^2 - 2ab * cos(C)
In our case, side XY is opposite to angle XZY, side YZ is opposite to angle YXZ, and side XZ is opposite to angle XYZ.
Given:
Side YZ = 13m
Angle XZY = -80 degrees (negative because it is clockwise)
Angle XYZ = 180 degrees - angle XZY - angle YXZ
Substituting the values into the Law of Cosines:
YZ^2 = XY^2 + XZ^2 - 2 * XY * XZ * cos(XYZ)
Solving for cos(XYZ):
cos(XYZ) = (XY^2 + XZ^2 - YZ^2) / (2 * XY * XZ)
Substitute the lengths:
cos(XYZ) = (XY^2 + XZ^2 - YZ^2) / (2 * XY * XZ)
cos(XYZ) = (13^2 + XZ^2 - XY^2) / (2 * 13 * XZ)
6. We have the value of cos(XYZ). Now, we need to find the angle XYZ (bearing of Z from Y). We can use inverse cosine (arccos) to find the angle.
XYZ = arccos(cos(XYZ))
Calculate arccos(XYZ) using a calculator or software:
XYZ = arccos[-((XY^2 + XZ^2 - YZ^2) / (2 * XY * XZ))]
Finally, we obtain the bearing of Z from Y, given the values of XY, XZ, and YZ.
Note: Make sure to check the units used for the angles (degrees/radians) and convert if necessary.