To analyze the coordinates \(A(6,3)\) and \(B(-3,1)\), we can calculate various properties and measures. Here are a few calculations:

### Distance Between Two Points
The distance \(d\) between two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) can be calculated using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

For \(A(6,3)\) and \(B(-3,1)\):
\[ d = \sqrt{(-3 - 6)^2 + (1 - 3)^2} \]
\[ d = \sqrt{(-9)^2 + (-2)^2} \]
\[ d = \sqrt{81 + 4} \]
\[ d = \sqrt{85} \]
\[ d = 9.22 \]

### Midpoint of Two Points
The midpoint \(M\) between two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

For \(A(6,3)\) and \(B(-3,1)\):
\[ M = \left( \frac{6 + (-3)}{2}, \frac{3 + 1}{2} \right) \]
\[ M = \left( \frac{3}{2}, \frac{4}{2} \right) \]
\[ M = \left( 1.5, 2 \right) \]

### Slope of the Line
The slope \(m\) of the line through two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

For \(A(6,3)\) and \(B(-3,1)\):
\[ m = \frac{1 - 3}{-3 - 6} \]
\[ m = \frac{-2}{-9} \]
\[ m = \frac{2}{9} \]

### Equation of the Line
The equation of the line in slope-intercept form \(y = mx + c\) can be determined using the slope \(m\) and a point \((x_1, y_1)\) on the line. We already have \(m = \frac{2}{9}\) and we can use point \(A(6,3)\):

\[ y - y_1 = m(x - x_1) \]
\[ y - 3 = \frac{2}{9}(x - 6) \]
\[ y - 3 = \frac{2}{9}x - \frac{12}{9} \]
\[ y = \frac{2}{9}x - \frac{4}{3} + 3 \]
\[ y = \frac{2}{9}x + \frac{5}{3} \]

Would you like any further details or have any questions about these calculations?

Here are 8 relative questions for deeper understanding:
1. How do you find the equation of a line passing through two given points?
2. What is the importance of the midpoint formula in coordinate geometry?
3. Can the distance formula be derived from the Pythagorean theorem? How?
4. How does the slope of a line affect its direction on a graph?
5. What are the applications of finding the distance between two points in real life?
6. How would you determine if two lines are parallel or perpendicular using their slopes?
7. What is the significance of the slope-intercept form of a line equation?
8. How can you find the area of a triangle formed by three points on a coordinate plane?

### Tip:
When dealing with coordinate geometry problems, always plot the points on a graph to visualize the problem better. This helps in understanding the relationships between the points and the lines formed.

Explore the analysis of coordinates A(6,3) and B(-3,1) in coordinate geometry, including calculating distance, midpoint, slope, and the equation of a line. Suitable for Grades 7-9.

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