A road runs from P to Q on a bearing of 140° and continues in the same direction. The bearings of R from P are 092° and 056° respectively, PQ is 1320 m. What is the shortest distance of R from the road?
To find the shortest distance of point R from the road, we need to use trigonometry. Here's how you can calculate it step by step:
Step 1: Draw a diagram
Start by drawing a diagram to visualize the given information. Label the points P, Q, and R, and draw a line segment PQ. Then draw lines from P to R, making angles of 92° and 56° with the line PQ.
Step 2: Find the angle PRQ
In triangle PRQ, we can find the angle PRQ by subtracting the given bearing of the road (140°) from 180° (since the sum of the angles in a triangle is 180°). Therefore, PRQ = 180° - 140° = 40°.
Step 3: Use the law of sines
Apply the law of sines to find the length of PR (the distance from R to the road). The law of sines states: a/sin(A) = b/sin(B) = c/sin(C), where a, b, c are the sides of the triangle, and A, B, C are their opposite angles.
In triangle PRQ, the side PR is opposite angle PRQ (40°), and the side PQ is opposite angle PQR (180° - 40° = 140°). The length of PQ is given as 1320 m.
Therefore, we have: PR/sin(40°) = PQ/sin(140°).
Step 4: Solve for PR
Rearrange the equation to solve for PR: PR = (PQ * sin(40°)) / sin(140°).
Using a scientific calculator or software, calculate sin(40°) and sin(140°).
PR = (1320 * sin(40°)) / sin(140°).
After evaluating this expression, you will find that PR is approximately 963.308 meters.
Therefore, the shortest distance of point R from the road is approximately 963.308 meters.