5.	For two points I and j on a plane surface, derive the observation for distance between them in plane coordinates. If the coordinates of point I are said to be fixed, what is the implication of this statement and reflect or show it in the observation equation for distance earlier derived. ( Hint: There is no guess attempt. Take distance ( ) i.e.  )

To derive the distance between two points \(I\) and \(J\) on a plane surface in plane coordinates, we use the Euclidean distance formula. Let's denote the coordinates of point \(I\) as \((x_1, y_1)\) and the coordinates of point \(J\) as \((x_2, y_2)\).

The distance \(d\) between points \(I\) and \(J\) is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Derivation

1. **Identify Coordinates:**
   - Point \(I\): \((x_1, y_1)\)
   - Point \(J\): \((x_2, y_2)\)

2. **Calculate the Differences in Coordinates:**
   - Difference in x-coordinates: \( \Delta x = x_2 - x_1 \)
   - Difference in y-coordinates: \( \Delta y = y_2 - y_1 \)

3. **Apply the Pythagorean Theorem:**
   - According to the Pythagorean theorem in a right triangle, the square of the hypotenuse (distance \(d\)) is the sum of the squares of the other two sides (differences in coordinates).
   - Thus, \( d^2 = (\Delta x)^2 + (\Delta y)^2 \)

4. **Take the Square Root:**
   - \( d = \sqrt{(\Delta x)^2 + (\Delta y)^2} \)
   - Substituting \(\Delta x\) and \(\Delta y\): \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

### Fixed Point \(I\)

If the coordinates of point \(I\) are fixed, it means that \(x_1\) and \(y_1\) are constants. This has the following implications for the observation equation:

1. **Fixed Coordinates:**
   - Since \(x_1\) and \(y_1\) are fixed, they do not change. Only the coordinates of point \(J\) (\(x_2\) and \(y_2\)) are variable.

2. **Simplified Observation Equation:**
   - The observation equation remains the same, but with \(x_1\) and \(y_1\) as constants: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
   - This implies that for any given point \(J\), the distance \(d\) is a function of the coordinates of \(J\) relative to the fixed point \(I\).

### Observation Equation for Fixed Point \(I\):

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Here, \(x_1\) and \(y_1\) are fixed values, and \(d\) is computed based on the coordinates \((x_2, y_2)\) of point \(J\).

Would you like further details or have any questions about this derivation?

Here are 5 more questions you might consider asking:
1. How do you derive the midpoint between two points in a plane?
2. What is the formula for the slope of the line passing through two points?
3. How do you find the equation of a line given two points?
4. Can you explain how to calculate the area of a triangle given its vertices in plane coordinates?
5. What is the distance formula in three-dimensional space?

**Tip:** Always double-check your coordinate differences before substituting into the distance formula to avoid sign errors.

Learn how to derive the distance between two points on a plane using the Euclidean distance formula. Understand the implications of fixed coordinates on the distance equation. Suitable for high school students and beyond.

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