Find the coordinate of the point p (x,y) that divide the line segment (A,B) with coordinate (5,3) and (1,3) in the ratio 3:1 internally and externally 

To find the coordinates of the point \( P(x, y) \) that divides the line segment joining two points \( A(5, 3) \) and \( B(1, 3) \) in a given ratio, we can use the section formula.

### Internal Division:
For internal division, the formula for the coordinates of the point \( P(x, y) \) that divides the line segment joining two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \) is:
\[
P(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\]
Given:
- \( A(5, 3) \)
- \( B(1, 3) \)
- Ratio \( m:n = 3:1 \)

Substitute the values:
\[
x = \frac{(3 \cdot 1) + (1 \cdot 5)}{3 + 1} = \frac{3 + 5}{4} = \frac{8}{4} = 2
\]
\[
y = \frac{(3 \cdot 3) + (1 \cdot 3)}{3 + 1} = \frac{9 + 3}{4} = \frac{12}{4} = 3
\]

So, the point \( P \) dividing the line segment internally in the ratio 3:1 is \( (2, 3) \).

### External Division:
For external division, the formula for the coordinates of the point \( P(x, y) \) that divides the line segment joining two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) externally in the ratio \( m:n \) is:
\[
P(x, y) = \left( \frac{mx_2 - nx_1}{m-n}, \frac{my_2 - ny_1}{m-n} \right)
\]

Substitute the values:
\[
x = \frac{(3 \cdot 1) - (1 \cdot 5)}{3 - 1} = \frac{3 - 5}{2} = \frac{-2}{2} = -1
\]
\[
y = \frac{(3 \cdot 3) - (1 \cdot 3)}{3 - 1} = \frac{9 - 3}{2} = \frac{6}{2} = 3
\]

So, the point \( P \) dividing the line segment externally in the ratio 3:1 is \( (-1, 3) \).

### Final Answer:
- The point dividing the line segment **internally** in the ratio 3:1 is \( (2, 3) \).
- The point dividing the line segment **externally** in the ratio 3:1 is \( (-1, 3) \).

Would you like more details on any part of this? Here are some related questions for further exploration:

1. How is the section formula derived?
2. Can you explain how the section formula changes for different types of divisions?
3. What if the points A and B had different y-coordinates instead of both being 3?
4. How can you apply the section formula in three-dimensional geometry?
5. How is the external division formula used in vector geometry?

**Tip:** When using the section formula, always ensure the ratio values are consistent (internal or external) and simplify the calculations carefully.

Find the coordinate of the point P (x,y) that divides the line segment (A,B) with coordinates (5,3) and (1,3) in the ratio 3:1 internally and externally.

Learn how to find the coordinates of the point that divides a line segment with endpoints (5,3) and (1,3) in the ratio 3:1 both internally and externally. This solution uses the section formula and is suitable for students in Grades 9-11.

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