QuestionAugust 26, 2025

7. Given overline (DF) with D(-1,11) and F(-9,-5) . if E partitions DF such that the ratio of DE to DF is 5:8 find the coordinates of E.

7. Given overline (DF) with D(-1,11) and F(-9,-5) . if E partitions DF such that the ratio of DE to DF is 5:8 find the coordinates of E.
7. Given overline (DF) with D(-1,11) and F(-9,-5) . if E partitions DF such that the ratio of DE to DF is 5:8 find the coordinates of E.

Solution
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Answer

The coordinates of E are (-6, 1). Explanation 1. Calculate the total distance DF Use the distance formula: DF = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}. Here, x_1 = -1, y_1 = 11, x_2 = -9, y_2 = -5. DF = \sqrt{(-9 + 1)^2 + (-5 - 11)^2} = \sqrt{(-8)^2 + (-16)^2} = \sqrt{64 + 256} = \sqrt{320}. 2. Determine the partition point E using the ratio The coordinates of E can be found using the section formula: E(x, y) = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) where m:n is the ratio. Here, m = 5, n = 3 (since DE:EF = 5:3), x_1 = -1, y_1 = 11, x_2 = -9, y_2 = -5. 3. Calculate the x-coordinate of E E_x = \frac{5(-9) + 3(-1)}{5+3} = \frac{-45 - 3}{8} = \frac{-48}{8} = -6. 4. Calculate the y-coordinate of E E_y = \frac{5(-5) + 3(11)}{5+3} = \frac{-25 + 33}{8} = \frac{8}{8} = 1.

Explanation

1. Calculate the total distance DF<br /> Use the distance formula: $DF = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. Here, $x_1 = -1$, $y_1 = 11$, $x_2 = -9$, $y_2 = -5$.<br /> $DF = \sqrt{(-9 + 1)^2 + (-5 - 11)^2} = \sqrt{(-8)^2 + (-16)^2} = \sqrt{64 + 256} = \sqrt{320}$.<br /><br />2. Determine the partition point E using the ratio<br /> The coordinates of E can be found using the section formula: $E(x, y) = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)$ where $m:n$ is the ratio.<br /> Here, $m = 5$, $n = 3$ (since $DE:EF = 5:3$), $x_1 = -1$, $y_1 = 11$, $x_2 = -9$, $y_2 = -5$.<br /><br />3. Calculate the x-coordinate of E<br /> $E_x = \frac{5(-9) + 3(-1)}{5+3} = \frac{-45 - 3}{8} = \frac{-48}{8} = -6$.<br /><br />4. Calculate the y-coordinate of E<br /> $E_y = \frac{5(-5) + 3(11)}{5+3} = \frac{-25 + 33}{8} = \frac{8}{8} = 1$.
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