Which points are closest to the origin?

Question

Certainly! Here's a rewritten version of your question:

"How can one determine which points on a Cartesian coordinate plane are closest to the origin, and what mathematical methods or formulas are used to calculate these distances?"

Accepted Answer

To determine which points on a Cartesian coordinate plane are closest to the origin, we need to calculate and compare the distances of each point from the origin. The origin in a 2D plane is represented by the coordinates (0, 0).

### Mathematical Method:

The distance between any point $(x, y)$ and the origin $(0, 0)$ can be calculated using the Euclidean distance formula. The Euclidean distance $d$ from the origin to a point $(x, y)$ is given by:

$$ d = \sqrt{x^2 + y^2} $$

### Steps to Determine Closest Points:

1. **Calculate the Distance for Each Point:**
   - For each point $(x_i, y_i)$ in the set of points you have, compute the distance to the origin using the formula:
   $$ d_i = \sqrt{x_i^2 + y_i^2} $$

2. **Compare Distances:**
   - After computing the distance for each point, compare all these calculated distances.

3. **Identify the Closest Point(s):**
   - The point(s) corresponding to the smallest distance(s) from the origin are the closest.

### Example:

Suppose you have the points $(3, 4)$, $(1, 1)$, and $(5, 6)$.

1. Calculate distances:
   - For $(3, 4)$:  
     $$ d_1 = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $$

- For $(1, 1)$:  
     $$ d_2 = \sqrt{1^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2} $$

- For $(5, 6)$:  
     $$ d_3 = \sqrt{5^2 + 6^2} = \sqrt{25 + 36} = \sqrt{61} $$

2. Compare distances:
   - $ \sqrt{2} \approx 1.414 $ (smallest)
   - $ 5 $
   - $ \sqrt{61} \approx 7.81 $

3. The point $(1, 1)$ is closest to the origin with a distance of $\sqrt{2}$.

### Conclusion:

By calculating and comparing the Euclidean distances of various points from the origin, you can determine which point(s) are the closest. In practice, sorting algorithms or optimization techniques might be employed for larger datasets to efficiently identify the closest points.

Which points are closest to the origin?

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