Question

1. 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

The formula to find the coordinates of a point \(Y(x,y)\) that divides the line - segment joining \(X(x_1,y_1)\) and \(Z(x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(m = 1\) and \(n=4\) (since \(Y\) divides \(XZ\) one - fifth of the way from \(X\) to \(Z\), the ratio of \(XY:YZ\) is \(1:4\)), \(x_1=-4,y_1 = 3,x_2 = 6,y_2=-2\).

Step2: Calculate the \(x\) - coordinate of \(Y\)

Substitute the values into the \(x\) - coordinate formula:
\[x=\frac{1\times6 + 4\times(-4)}{1 + 4}=\frac{6-16}{5}=\frac{-10}{5}=-2\]

Step3: Calculate the \(y\) - coordinate of \(Y\)

Substitute the values into the \(y\) - coordinate formula:
\[y=\frac{1\times(-2)+4\times3}{1 + 4}=\frac{-2 + 12}{5}=\frac{10}{5}=2\]

Answer:

\((-2,2)\)