Question

1. given ac with a(3, 4) and c(-4, -2), b partitions ac such that the ratio of ab to bc is 1:5. find the coordinates of b.
2. 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.
3. given $overline{xz}$ with x(-4, 3) and z(6, -2), find the coordinates of y if y divides xz one - fifth of the way from x to z.

Explanation:

Step1: Recall the section - formula

If a point \(P(x,y)\) divides the line - segment joining \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the ratio \(m:n\), then \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\).

Step2: Solve for question 7

Given \(D(-1,11)\) and \(F(-9,-5)\) and the ratio \(DE:DF = 5:8\), so \(m = 5\) and \(n=8 - 5=3\).
For the \(x\) - coordinate of \(E\):

$$x=\frac{5\times(-9)+3\times(-1)}{5 + 3}=\frac{-45-3}{8}=\frac{-48}{8}=-6$$

For the \(y\) - coordinate of \(E\):

$$y=\frac{5\times(-5)+3\times11}{5 + 3}=\frac{-25 + 33}{8}=\frac{8}{8}=1$$

So the coordinates of \(E\) are \((-6,1)\).

Step3: Solve for question 9

Given \(X(-4,3)\) and \(Z(6,-2)\) and \(Y\) divides \(XZ\) one - fifth of the way from \(X\) to \(Z\), so \(m = 1\) and \(n = 4\).
For the \(x\) - coordinate of \(Y\):

$$x=\frac{1\times6+4\times(-4)}{1 + 4}=\frac{6-16}{5}=\frac{-10}{5}=-2$$

For the \(y\) - coordinate of \(Y\):

$$y=\frac{1\times(-2)+4\times3}{1 + 4}=\frac{-2 + 12}{5}=\frac{10}{5}=2$$

So the coordinates of \(Y\) are \((-2,2)\).

Answer:

1. The coordinates of \(E\) are \((-6,1)\).
2. The coordinates of \(Y\) are \((-2,2)\).