**Correct Option: A. 5 kilometers** ### Step-by-Step Explanation 1. **Understanding Displacement**: Displacement is defined as the shortest distance from the initial position to the final position, along with the direction. It is a vector quantity, meaning it has both magnitude and direction. 2. **Visualizing the Path**: - The hiker starts at point A and walks 3 kilometers east to point B. - From point B, the hiker then walks 4 kilometers north to point C. - To visualize this, you can imagine a right triangle where: - The horizontal leg (eastward movement) is 3 kilometers. - The vertical leg (northward movement) is 4 kilometers. 3. **Using the Pythagorean Theorem**: - To find the displacement (the hypotenuse of the right triangle formed), we can use the Pythagorean theorem, which states: \[ c^2 = a^2 + b^2 \] where \(c\) is the hypotenuse (displacement), and \(a\) and \(b\) are the other two sides of the triangle. - In this case: - \(a = 3\) kilometers (east) - \(b = 4\) kilometers (north) 4. **Calculating the Displacement**: - Plugging in the values: \[ c^2 = 3^2 + 4^2 \] \[ c^2 = 9 + 16 \] \[ c^2 = 25 \] \[ c = \sqrt{25} = 5 \text{ kilometers} \] - Therefore, the displacement from point A to point C is 5 kilometers. 5. **Direction of Displacement**: - The direction of the displacement can be found using trigonometry. The angle θ with respect to the eastward direction can be calculated using: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{3} \] - This means the hiker's displacement is 5 kilometers at an angle θ north of east. ### Why Other Options Are Incorrect - **Option B (7 kilometers)**: This option might confuse students who think of the total distance traveled (3 km + 4 km = 7 km). However, total distance is not the same as displacement, which is the straight-line distance from the starting point to the endpoint. - **Option C (12 kilometers)**: This option is incorrect for the same reason as option B. It represents a misunderstanding of the concept of displacement versus total distance traveled. - **Option D (1 kilometer)**: This option is incorrect as it significantly underestimates the actual displacement. It does not relate to the distances traveled in any way. ### Common Pitfalls - Confusing total distance traveled with displacement. Remember, displacement is the straight-line distance between two points, not the sum of the distances traveled. - Misapplying the Pythagorean theorem by not correctly identifying the legs of the triangle formed by the path taken. ### Revision Summary - Displacement is the shortest distance from the starting point to the endpoint, measured in a straight line. - Use the Pythagorean theorem to calculate displacement in right triangles. - The correct displacement in this scenario is 5 kilometers. - Always differentiate between total distance traveled and displacement to avoid confusion.