Three police posts X, Y and Z are such that Y is 50km on a bearing 060 degrees from X while Z is 70km from Y and on a bearing of 300 degrees from X.
Determine the distance, in km, of Z and X.
Draw triangle XYZ, with sides x,y,z in the usual places, ooposite the angles.
That is, XZ = y
Using the law of sines,
z/sinZ = x/sinX
50/sinZ = 70/sin120°
Now, knowing X and Z, it's easy to find Y, and you can use either law of sines or law of cosines to find XZ = y
All angles are measured CW from +y-axis
XZ = XY + YZ = 50[60o] + 70[300o].
XZ = (50*sin60+70*sin300) + (50*cos60+70*cos300)I,
XZ = -17.3 + 60i = 62.4km[-16o] = 62.4km[344o] CW.
To determine the distance between point Z and point X, we can use the concept of vector addition.
First, let's find the coordinates of the three points:
- Point X: We start from a reference point (let's say, the origin) and move 50 km at a bearing of 060 degrees. Using basic trigonometry, we can find the x and y coordinates of point X:
- x-coordinate of point X = 50 km * cos(60°) = 25 km
- y-coordinate of point X = 50 km * sin(60°) = 43.3 km
- Point Y: Point Y is 50 km on a bearing of 060 degrees from point X, which means it is just 50 km in the same direction. So, the coordinates of point Y are the same as point X: (25 km, 43.3 km).
- Point Z: We start from point Y and move 70 km on a bearing of 300 degrees. Again, using trigonometry, we can find the x and y coordinates of point Z:
- x-coordinate of point Z = 70 km * cos(300°) = 35 km
- y-coordinate of point Z = 70 km * sin(300°) = -40.5 km (since the bearing is 300 degrees, which corresponds to the fourth quadrant)
Now, using the coordinates of point X (25 km, 43.3 km) and point Z (35 km, -40.5 km), we can find the distance between them using the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Distance = √((35 km - 25 km)^2 + (-40.5 km - 43.3 km)^2)
Distance = √((10 km)^2 + (-83.8 km)^2)
Distance = √(100 km^2 + 7006.44 km^2)
Distance = √(7106.44 km^2)
Distance ≈ 84.3 km
So, the distance between point Z and point X is approximately 84.3 kilometers.