Math Problem Statement

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

Solution

To find the coordinates of point FF, which is 23\frac{2}{3} of the way from E(3,1)E(-3, -1) to C(0,4)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 (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) in the ratio m:nm:n is:

(x,y)=(mx2+nx1m+n,my2+ny1m+n)(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)

Here, m:n=2:1m:n = 2:1 (since the ratio is 23\frac{2}{3}, the point divides the line in the ratio 2:1), (x1,y1)=(3,1)(x_1, y_1) = (-3, -1), and (x2,y2)=(0,4)(x_2, y_2) = (0, -4).

Now we can calculate the coordinates of point FF.

For the x-coordinate:

x=2(0)+1(3)2+1=033=33=1x = \frac{2(0) + 1(-3)}{2+1} = \frac{0 - 3}{3} = \frac{-3}{3} = -1

For the y-coordinate:

y=2(4)+1(1)2+1=813=93=3y = \frac{2(-4) + 1(-1)}{2+1} = \frac{-8 - 1}{3} = \frac{-9}{3} = -3

So, the coordinates of point FF are (1,3)(-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:nm:n is reversed?
  4. How would you find the equation of a line passing through points EE and CC?
  5. How can we verify that FF lies on the line between EE and CC?

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

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Section Formula
Ratios

Formulas

Section formula for internal division: (x, y) = ((mx2 + nx1) / (m + n), (my2 + ny1) / (m + n))

Theorems

Section formula for dividing a line segment in a given ratio

Suitable Grade Level

Grades 9-11