Point F, which is 2/3 the distance from E to C. The coordinate of E is -3,-1 and C is 0,-4. So what is the coordinates of point F

To find the coordinates of point \( F \), which is \(\frac{2}{3}\) of the way from \( E(-3, -1) \) to \( C(0, -4) \), we can use the section formula for internal division of a line segment in a given ratio.

The formula for the coordinates of a point dividing a line segment between \( (x_1, y_1) \) and \( (x_2, y_2) \) in the ratio \( m:n \) is:

\[
(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\]

Here, \( m:n = 2:1 \) (since the ratio is \(\frac{2}{3}\), the point divides the line in the ratio 2:1), \( (x_1, y_1) = (-3, -1) \), and \( (x_2, y_2) = (0, -4) \).

Now we can calculate the coordinates of point \( F \).

### For the x-coordinate:
\[
x = \frac{2(0) + 1(-3)}{2+1} = \frac{0 - 3}{3} = \frac{-3}{3} = -1
\]

### For the y-coordinate:
\[
y = \frac{2(-4) + 1(-1)}{2+1} = \frac{-8 - 1}{3} = \frac{-9}{3} = -3
\]

So, the coordinates of point \( F \) are \( (-1, -3) \).

Would you like further details or have any questions?

Here are some related questions to explore:
1. How do you calculate the midpoint between two points?
2. Can you apply the section formula to divide a line externally?
3. What happens when the ratio \( m:n \) is reversed?
4. How would you find the equation of a line passing through points \( E \) and \( C \)?
5. How can we verify that \( F \) lies on the line between \( E \) and \( C \)?

**Tip:** The section formula can be used for both internal and external division of a line segment.

Point F, which is 2/3 the distance from E to C. The coordinate of E is -3,-1 and C is 0,-4. So what is the coordinates of point F?

Learn how to calculate the coordinates of point F, which is 2/3 of the way from E(-3,-1) to C(0,-4). This problem uses the section formula for internal division and is suitable for students in Grades 9-11.

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