Bearings | Distance Bearing Problems | Likely Examination Questions | Maths Center
TLDRThe video script presents a detailed mathematical problem involving a ship's journey from Port R to Port S and then to Port K. The ship first sails 54 kilometers on a bearing of 65 degrees from Port R to Port S, then continues for 80 kilometers on a bearing of 155 degrees to reach Port K. The challenge is to calculate the direct distance between Port R and Port K, as well as the bearing of Port K from Port R. The solution involves drawing a diagram to visualize the journey, forming right-angle triangles, and applying trigonometric principles and the Pythagorean theorem. The distance between Port R and Port K is found to be 96.5 kilometers, and the bearing of Port K from Port R is determined to be 121 degrees, rounding to one decimal place.
Takeaways
- 📐 **Understanding the Problem**: Carefully read the problem statement and draw a diagram to visualize the journey from port R to S and then to port K.
- 🧭 **Bearing Calculation**: The ship's journey from port R to S is on a bearing of 65 degrees, while the journey from port S to K is on a bearing of 155 degrees.
- 📏 **Distance Measurement**: The distance from port R to S is given as 54 kilometers, and from port S to K is 80 kilometers.
- 🔍 **Geometry Application**: Use geometric principles to form right-angle triangles and find interior angles, which are essential for solving the problem.
- 🔢 **Pythagorean Theorem**: Apply the Pythagorean theorem to find the distance between port R and K (RK), knowing the lengths of RS and SK.
- 📐 **Angle Calculation**: Calculate the unknown angle (theta) in the right triangle RSK using trigonometric ratios, specifically the tangent function.
- 🔢 **Trigonometric Ratios**: Utilize the tangent ratio (opposite/adjacent) to find the value of angle theta, which is crucial for determining the bearing from R to K.
- ➗ **Bearing Calculation**: The final bearing of K from R is found by adding the known bearing angle (65 degrees) to the calculated angle theta (56 degrees).
- 📈 **Approximation**: Round the final bearing to one decimal place, resulting in a bearing of 121 degrees from R to K.
- 📝 **Accuracy in Diagram**: Ensure the diagram accurately represents the lengths and angles, as this affects the correctness of the calculated distances and bearings.
- 🤔 **Problem-Solving Approach**: The process involves understanding the problem, drawing a diagram, applying geometric and trigonometric principles, and performing calculations to find the required distances and bearings.
Q & A
What is the initial bearing from Port R to Port S?
-The initial bearing from Port R to Port S is 65 degrees.
How far does the ship sail from Port R to Port S?
-The ship sails a distance of 54 kilometers from Port R to Port S.
What bearing does the ship take from Port S to Port K?
-The ship takes a bearing of 155 degrees from Port S to Port K.
How far does the ship sail from Port S to Port K?
-The ship sails a distance of 80 kilometers from Port S to Port K.
How is the distance between Port R and Port K calculated?
-The distance between Port R and Port K is calculated using the Pythagorean theorem, considering the right-angled triangle RSK, where RS is 54 km and SK is 80 km.
What is the distance between Port R and Port K?
-The distance between Port R and Port K is 96.5 kilometers.
What is the angle formed at Port S in the triangle RSK?
-The angle formed at Port S in the triangle RSK is 25 degrees, which is calculated by subtracting 65 degrees from 90 degrees.
How are alternating angles used in this problem?
-Alternating angles are used to establish that the angle at Port R is also 65 degrees since the line RS is a transversal crossing two parallel lines (power lines).
What trigonometric ratio is used to find the angle theta?
-The tangent ratio (tan) is used to find the angle theta, as it relates the opposite side to the adjacent side in a right-angled triangle.
How is the bearing of Port K from Port R determined?
-The bearing of Port K from Port R is determined by adding the angle theta to the initial bearing from Port R to Port S, which is 65 degrees.
What is the final bearing of Port K from Port R?
-The final bearing of Port K from Port R is 121 degrees, after adding the angle theta (56 degrees) to the initial bearing of 65 degrees.
Why is it important to draw an accurate diagram when solving this problem?
-Drawing an accurate diagram is crucial as it helps visualize the problem, establish the correct angles and distances, and ensures that the calculations are based on a correct representation of the given information.
Outlines
🚢 Sailing Path and Diagram Construction
The video begins with an introduction to the problem of a ship's journey from port R to port S on a 65-degree bearing, covering a distance of 54 kilometers, followed by a second leg from port S to port K on a 155-degree bearing, covering 80 kilometers. The first step in solving this problem is to carefully read the statements and draw a diagram to visualize the ship's path. The diagram includes the starting point at port R, the bearing to port S, and the subsequent bearing from port S to port K. The importance of accurately representing distances and bearings in the diagram is emphasized to ensure the correct solution.
📐 Calculating the Interior Angles and Triangle Formation
The second paragraph focuses on forming right-angle triangles from the given information to apply geometric principles. It explains the process of finding the interior angles of the triangle formed by the ship's path. By using the properties of parallel lines (power lines) and the concept of alternating angles, the angles at port S are determined to be 65 degrees and 25 degrees. This allows the formation of a right-angled triangle (triangle RSK), which is crucial for the subsequent application of the Pythagorean theorem.
🔢 Applying Pythagoras' Theorem to Find Distance
In this paragraph, the Pythagorean theorem is applied to the right-angled triangle RSK to find the distance between port R and port K. With the distances between S and K (80 km) and S and R (54 km) known, the theorem is used to calculate the hypotenuse RK. By substituting the known values into the theorem, the distance RK is found to be 96.5 kilometers after taking the square root of the sum of the squares of the other two sides.
🧭 Determining the Bearing of Port K from Port R
The final paragraph addresses the calculation of the bearing of port K from port R. It introduces the concept of using trigonometric ratios to find the unknown angle theta, which is necessary to determine the bearing. By applying the tangent function to the opposite and adjacent sides of triangle RSK, theta is calculated to be approximately 56 degrees. Adding this to the known 65 degrees bearing from R to S gives the total bearing from port R to port K as 121 degrees, rounded to one decimal place as per the instructions in the problem.
Mindmap
Keywords
💡Bearing
💡Port R
💡Port S
💡Port K
💡Distance
💡Pythagorean Theorem
💡Trigonometry
💡Right-Angle Triangle
💡Transversal
💡Alternating Angles
💡Tangent
💡Approximation
Highlights
A ship sails from port R on a bearing of 65 degrees to port S, a distance of 54 kilometers.
The ship then sails on a bearing of 155 degrees from port S to port K, a distance of 80 kilometers.
The first step is to read the statements carefully and draw an appropriate diagram for the given information.
The diagram illustrates the ship's journey from port R to S and then from S to K.
The distance between port R and port S is 54 kilometers, and between port S and port K is 80 kilometers.
To find the distance between R and K, form right-angle triangles from the given information.
Identify the interior angles of the triangle using geometry to apply the Pythagorean theorem.
The right-angled triangle is formed with sides RS and SK, with SR being the hypotenuse.
Apply the Pythagorean theorem: RS^2 = RK^2 + SK^2, where RK is the distance to be found.
Calculate the square of the sides (54 km and 80 km) and solve for the unknown side RK.
The distance between port R and port K (RK) is found to be 96.5 kilometers using the Pythagorean theorem.
To find the bearing of K from R, measure the angle from the zero line of R to the line joining R to K.
Use trigonometric ratios to find the unknown angle theta, which is part of the bearing calculation.
Theta is calculated using the tangent function, where theta = tan^(-1)(opposite/adjacent).
The bearing of K from R is found by adding the angle theta to the known angle of 65 degrees.
The final bearing of K from R is calculated to be 121 degrees, rounded to one decimal place.
The problem-solving approach involves careful reading, diagramming, geometric analysis, and trigonometric calculations.
The method demonstrates the practical application of trigonometry and geometry in navigation and distance problems.
Transcripts
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