House B is located at a bearing of N67°E from house A. House C is 370 meters from house A at a bearing of S68°E. House B is located at a bearing of N11°W from house C. Find the distance from house A to house B. (Round your answer to two decimal places.)
Sorry, I would post my drawing but it won't let me!
Extend AB to meet the vertical from C at D.
It is easy to determine that in triangle ABC
∠C = 47°
∠B = 88°
Now just use the law of sines to get AB.
No worries! I can help you solve this problem without the need for a drawing. Let's break it down step by step.
1. Start by identifying the bearings and distances given in the problem:
- House B is located at a bearing of N67°E from house A.
- House C is 370 meters from house A at a bearing of S68°E.
- House B is located at a bearing of N11°W from house C.
2. To solve this problem, we'll use the concept of relative bearings. The relative bearing is the angle between the reference line (in this case, North) and the line connecting two points.
3. Firstly, let's find the relative bearing of house B with respect to house A. We know that house B is located at a bearing of N67°E from house A. To get the relative bearing, subtract the given bearing from 90°:
- Relative bearing of B from A = 90° - 67° = 23°
4. Next, we'll find the relative bearing of house C with respect to house A. House C is located at a bearing of S68°E from house A. To convert the bearing into a relative bearing, subtract it from 180°:
- Relative bearing of C from A = 180° - 68° = 112°
5. Now, let's find the relative bearing of house B with respect to house C. We know that house B is located at a bearing of N11°W from house C. To convert the bearing into a relative bearing, add it to 180°:
- Relative bearing of B from C = 180° + 11° = 191°
6. We can now create a triangle using the three points: A, B, and C. The side opposite the angle with a relative bearing of 23° is the distance between houses A and B. The side opposite the angle with a relative bearing of 112° is the distance between houses A and C. We need to find the length of the side opposite the angle with a relative bearing of 191°, which represents the distance from house A to house B.
7. To calculate the distance from house A to house B, we can use the Law of Sines. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the opposite angle is constant for all three sides and angles. In this case, we can use the following formula:
- Distance from A to B / sin(112°) = Distance from A to C / sin(23°)
8. Rearranging the formula, we get:
- Distance from A to B = (Distance from A to C * sin(112°)) / sin(23°)
9. Substitute the given distance from A to C, which is 370 meters, into the formula:
- Distance from A to B = (370 * sin(112°)) / sin(23°)
10. Using a scientific calculator or online trigonometric calculator, calculate the value of sin(112°) and sin(23°).
11. Finally, plug in the calculated values and solve for the distance from A to B.
Note: The final answer should be rounded to two decimal places, as specified in the question.