Q, P and R are three villages on a level ground. Q is 4km on a bearing of 040° from P, while R is 3km on a bearing of 130° from Q. Calculate the distance and bearing of P from R.
Draw the diagram. It is clear that angle PQR is 90° in a 3-4-5 triangle. So, PR = 5km
As for the direction, you can either figure the x- and y-coordinates of R, or note that angle QRP is arcsin(4/5), and add that to the diagram.
To calculate the distance and bearing of P from R, we can break down the problem into steps:
Step 1: Determine the position of Q with respect to P.
Step 2: Determine the position of R with respect to Q.
Step 3: Add the distances and bearings to find the position of R with respect to P.
Step 1: Determine the position of Q with respect to P.
Q is 4km on a bearing of 040° from P. This means that if we draw a straight line from P to Q, it will have a length of 4km and an angle of 040°.
Step 2: Determine the position of R with respect to Q.
R is 3km on a bearing of 130° from Q. This means that if we draw a straight line from Q to R, it will have a length of 3km and an angle of 130°.
Step 3: Add the distances and bearings to find the position of R with respect to P.
To find the position of R with respect to P, we need to combine the distances and bearings from Steps 1 and 2.
First, let's determine the position of Q with respect to P. Since Q is 4km on a bearing of 040° from P, we can draw a straight line from P to Q with a length of 4km and an angle of 040°.
Next, let's determine the position of R with respect to Q. Since R is 3km on a bearing of 130° from Q, we can draw a straight line from Q to R with a length of 3km and an angle of 130°.
Now, if we draw a straight line from P to R, it will pass through Q. We can treat the three lines (from P to Q, from Q to R, and from P to R) as a triangle. To find the distance and bearing of P from R, we can use the law of cosines and sine rule:
Using the law of cosines, we know that:
c^2 = a^2 + b^2 - 2ab * cos(C)
where c is the distance between P and R, a is the distance between P and Q, b is the distance between Q and R, and C is the angle between a and b (which can be found by subtracting the angles of a and b from 180).
In this case, a = 4km, b = 3km, and cos(C) can be found using the given bearings.
We can use the sine rule to find the bearing of P from R:
sin(A) / a = sin(C) / c
where A is the bearing of P from R, c is the distance between P and R, and C is the angle between a and c (which can be found by subtracting the angles of b and a from 180).
Using these formulas, we can calculate the distance and bearing of P from R.
To calculate the distance and bearing of P from R, we can use vector addition and trigonometry.
Step 1: Plot the villages on a diagram. Let's assume P is our origin (0,0) on a coordinate plane. Q is located 4km away at a bearing of 040°, and R is located 3km away at a bearing of 130° from Q.
Step 2: Find the coordinates of Q. Since Q is 4km away at a bearing of 040° from P, we need to find the x and y coordinates of Q:
x-coordinate of Q = 4 km * sin(040°)
y-coordinate of Q = 4 km * cos(040°)
Step 3: Find the coordinates of R. Since R is 3km away at a bearing of 130° from Q, we need to find the x and y coordinates of R:
x-coordinate of R = x-coordinate of Q + 3 km * sin(130°)
y-coordinate of R = y-coordinate of Q + 3 km * cos(130°)
Step 4: Calculate the distance between P and R using the coordinates:
Distance = sqrt((x-coordinate of R)^2 + (y-coordinate of R)^2)
Step 5: Calculate the bearing of P from R. We need to find the angle between the x-axis and the line connecting P and R:
Bearing = 180° + atan2(y-coordinate of R, x-coordinate of R)
Plug in the values and calculate the answer.