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Generate an image illustrating an abstract concept of a triangle formed by three points, marked as P, Q, and R. Point P should be placed up and should be perceived to be towards the north, Q should be placed at the bottom and recognized as the south, and R should be to the southeast of P, indicating the bearings. The lines linking these points should be symbolically represented as sticks, neatly arranged with no text included. The total distance should be represented with a dashed line from P to R, though the actual measurement isn't specified in this depiction.

P, q, r are arranged sticks. P is 10km north of q. The bearing of r from p is 223, and the bearing of q from r is 097. Find the distance of r from p. Show full working both the bearing resulting diagram and the formular

Draw a diagram. The measures of the angles of triangle PQR are

P:43
Q:83
r:54

Now just use the law of sines to get PR:

PR/sin83 = 10/sin54

To find the distance of r from p, we can use the sine rule along with the information about the bearings:

Let's assume the distance of r from p is x km.

We know that the bearing of r from p is 223 degrees, which means we can draw a triangle with p as the starting point and r as the end point, and the bearing from p to r is 223 degrees. Similarly, the bearing of q from r is 97 degrees.

Now, let's denote angle pqr as A, angle rqp as B, and angle rpq as C.

Using the given information, we have:
Angle A = 223 degrees
Angle B = 97 degrees
Angle C = 360 - (A + B) = 40 degrees

According to the sine rule:
a/sinA = b/sinB = c/sinC

Let's set up the equation using this rule:
pq / sinA = qr / sinC

Substituting the given values:
10 / sin(223) = x / sin(40)

Taking the sin inverse of both sides:
x = (10 * sin(40)) / sin(223)

Using a scientific calculator, we can find the value of x to be approximately 10.35 km.

Therefore, the distance of r from p is approximately 10.35 km.

To solve this problem, we can use the concept of bearings and the trigonometric functions of sine and cosine. Here's a step-by-step explanation:

1. First, let's draw a diagram to visualize the situation. Draw three points P, Q, and R. Label P as 10km north of Q. Now, we need to determine the bearings of R from P and Q from R.

P --------------> Q
|
|
R

2. According to the problem, the bearing of R from P is 223. This means that the angle formed between the line joining P and R and the north direction is 223 degrees. Mark this angle on the diagram.

P --------------> Q
\ |
\ |
\ |
\ R (223°)

3. Similarly, the bearing of Q from R is given as 097. This means that the angle formed between the line joining R and Q and the north direction is 97 degrees. Mark this angle on the diagram.

P --------------> Q (097°)
\ |
\ |
\ |
\ R (223°)

4. We need to find the distance between P and R. To do this, we can use the concept of trigonometry. In the right triangle formed by P, R, and the vertical line (north direction), we will use the sine function. The formula is: sin(angle) = opposite/hypotenuse.

5. Let's calculate the opposite side of the triangle, which is the distance between R and Q. We can use the formula: sin(97°) = opposite/10km (because P is 10km north of Q).

sin(97°) = opposite/10km
opposite = sin(97°) * 10km

6. Now, let's calculate the hypotenuse of the triangle, which is the distance between P and R. Using the formula: sin(223°) = opposite/hypotenuse.

sin(223°) = opposite/hypotenuse
hypotenuse = opposite/sin(223°)

7. Substitute the value of opposite (calculated in step 5) into the hypotenuse equation:

hypotenuse = (sin(97°) * 10km) / sin(223°)

8. Calculate the value of hypotenuse using a calculator. The result will give you the distance between P and R in kilometers.

By following these steps, you will be able to find the distance of R from P using the given bearings and trigonometry formulas.