Question

1. given $overline{ac}$ with $a(3,4)$ and $c(-9,-2)$, if $b$ partitions $overline{ac}$ such that the ratio of $ab$ to $bc$ is $1:5$, find the coordinates of $b$.

Explanation:

Step1: Recall the section - formula

If a point \(B(x,y)\) divides the line - segment joining \(A(x_1,y_1)\) and \(C(x_2,y_2)\) in the ratio \(m:n\), then \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, the ratio of \(AB\) to \(BC\) is \(1:5\), so \(m = 1\) and \(n = 5\), \(x_1=3,y_1 = 4,x_2=-9,y_2=-2\).

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

Using the formula \(x=\frac{mx_2+nx_1}{m + n}\), substitute \(m = 1\), \(n = 5\), \(x_1=3\), and \(x_2=-9\).
\[x=\frac{1\times(-9)+5\times3}{1 + 5}=\frac{-9 + 15}{6}=\frac{6}{6}=1\]

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

Using the formula \(y=\frac{my_2+ny_1}{m + n}\), substitute \(m = 1\), \(n = 5\), \(y_1 = 4\), and \(y_2=-2\).
\[y=\frac{1\times(-2)+5\times4}{1 + 5}=\frac{-2+20}{6}=\frac{18}{6}=3\]

Answer:

The coordinates of \(B\) are \((1,3)\)