To find the coordinates of point F that divides the line segment DE in the ratio of DF:EF = 2:3, we can use the section formula. The section formula states that if a point F divides the line segment joining points D(x1, y1) and E(x2, y2) in the ratio m:n, then the coordinates of F can be given by:
\[ F\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
In this case:
- \(D(-7, -8)\) indicates \(x_1 = -7\) and \(y_1 = -8\),
- \(E(3, 7)\) indicates \(x_2 = 3\) and \(y_2 = 7\),
- The ratio m:n = 2:3, so \(m = 2\) and \(n = 3\).
Now, substituting the values into the section formula:
- Calculate the x-coordinate of F:
\[ x_F = \frac{2 \cdot 3 + 3 \cdot (-7)}{2 + 3} = \frac{6 - 21}{5} = \frac{-15}{5} = -3 \]
- Calculate the y-coordinate of F:
\[ y_F = \frac{2 \cdot 7 + 3 \cdot (-8)}{2 + 3} = \frac{14 - 24}{5} = \frac{-10}{5} = -2 \]
Therefore, the coordinates of point F are:
\[ F(-3, -2) \]
So, the coordinates of F are \((-3, -2)\).
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