Question

1. given $overline{wy}$ with $w(3,7)$ and $y(13, - 8)$, if $x$ partitions $overline{wy}$ such that the ratio of $wx$ to $xy$ is $3:2$, find the coordinates of $x$.

Explanation:

Step1: Recall section - formula

The section - formula for a point \(P(x,y)\) that divides the line - segment joining \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the ratio \(m:n\) is given by \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, the ratio of \(WX\) to \(XY\) is \(3:2\), so the point \(X\) divides the line - segment \(\overline{WY}\) in the ratio \(3:2\), where \(W(x_1,y_1)=(3,7)\) and \(Y(x_2,y_2)=(13, - 8)\), and \(m = 3\), \(n = 2\).

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

Using the formula \(x=\frac{mx_2+nx_1}{m + n}\), we substitute \(x_1 = 3\), \(x_2=13\), \(m = 3\), and \(n = 2\).
\[

$$\begin{align*} x&=\frac{3\times13+2\times3}{3 + 2}\\ &=\frac{39 + 6}{5}\\ &=\frac{45}{5}\\ &=9 \end{align*}$$

\]

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

Using the formula \(y=\frac{my_2+ny_1}{m + n}\), we substitute \(y_1 = 7\), \(y_2=-8\), \(m = 3\), and \(n = 2\).
\[

$$\begin{align*} y&=\frac{3\times(-8)+2\times7}{3 + 2}\\ &=\frac{-24 + 14}{5}\\ &=\frac{-10}{5}\\ &=-2 \end{align*}$$

\]

Answer:

The coordinates of \(X\) are \((9,-2)\)