A man walks 100m due West from point X to Y, he then walks 100m due North to point Z. Find the bearing of X from Z
draw the lines. the man walked W45°N, so
the bearing is S45°E or E45°S
change that to 0-360° if you want.
Well, since the man walked 100m due West and then 100m due North, we can say that he took a detour just to avoid asking for directions! But let's find the bearing of X from Z.
To find the bearing, we need to use trigonometry. The distance between X and Z forms a right-angled triangle. So, let's use the tangent function.
The tangent of an angle is equal to the ratio of the opposite side to the adjacent side in a right-angled triangle. In this case, the opposite side is 100m, and the adjacent side is 100m.
So, the tangent of the angle formed by X, Z, and the horizontal is 100m/100m, which is equal to 1.
Using inverse tangent (arctan), we find that the angle is 45 degrees.
Now here comes the funny part – 45 degrees is halfway between North and West. So, if we were to convert it to a bearing, we could say the man is a bit lost in a magical land called "Northwest-ish"!
But if we really have to assign a specific bearing, we can say the bearing of X from Z is approximately 315 degrees, which is northwest by the traditional compass. Just be careful not to take a detour there!
To find the bearing of point X from point Z, we can use trigonometry.
Step 1: Draw a diagram to represent the situation. Place point X to the left of point Z, with a distance of 100m between them.
Step 2: Draw a right-angled triangle using points X, Y, and Z. The line connecting X and Z is the hypotenuse of this triangle.
Step 3: Label the angle between the line connecting X and Z and the horizontal line as angle A.
Step 4: We know that the man initially walks 100m due West from X to Y, which is the adjacent side of angle A. And then, he walks 100m due North from Y to Z, which is the opposite side of angle A.
Step 5: We can use the inverse tangent function (arctan) to find the measure of angle A, which is the bearing of X from Z. The formula is:
tan(A) = opposite / adjacent
So, tan(A) = 100m / 100m = 1
Step 6: Taking the arctan of both sides of the equation, we have:
A = arctan(1)
Step 7: Using a calculator, the value of A is approximately 45 degrees.
Step 8: Therefore, the bearing of X from Z is 45 degrees.
To find the bearing of X from Z, we need to determine the angle between the line connecting X and Z and the North direction.
First, we can visualize the scenario. Let's assume that X is our starting point and Z is the destination. Given that the man walks 100m due West from X to Y, and then 100m due North from Y to Z, we can draw a right-angled triangle with sides XY, YZ, and XZ.
We can label the right-angle at Y as angle θ, which represents the bearing of X from Z.
To find θ, we can use trigonometry.
Since the man walks 100m due West from X to Y, the length of XY is 100m.
Then, since the man walks 100m due North from Y to Z, the length of YZ is also 100m.
Using the Pythagorean theorem, we can calculate the length of XZ (the hypotenuse of the right triangle):
XZ² = XY² + YZ²
XZ² = 100² + 100²
XZ² = 20000 + 20000
XZ² = 40000
XZ = √40000
XZ = 200√2
Now that we know the lengths of all the sides of the triangle, we can find the sine and cosine of angle θ:
sin(θ) = YZ / XZ
sin(θ) = 100 / (200√2)
sin(θ) = 1 / (2√2)
sin(θ) = √2 / 4
cos(θ) = XY / XZ
cos(θ) = 100 / (200√2)
cos(θ) = 1 / (2√2)
cos(θ) = √2 / 4
Finally, we can find θ by taking the arctan of the ratio of sine to cosine:
θ = atan(sin(θ) / cos(θ))
θ = atan((√2 / 4) / (√2 / 4))
θ = atan(1)
θ ≈ 45 degrees
Therefore, the bearing of X from Z is approximately 45 degrees.