Question

1. given $overline{ac}$ with $a(3,4)$ and $c(-9,-2)$. if $b$ 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 $ef$ is $5:8$, find the coordinates of $e$.
3. given $overline{xz}$ with $x(-4,3)$ and $z(6,-3)$. find the coordinates of $y$ if $y$ divides $overline{xz}$ one - third of the way from $x$ to $z$

Explanation:

Step1: Recall 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 \(y=\frac{my_2+ny_1}{m + n}\).

Step2: Identify values for problem 5

For the line - segment \(AC\) with \(A(3,4)\) and \(C(-9,-2)\) and the ratio \(m:n = 1:5\), \(x_1 = 3\), \(y_1 = 4\), \(x_2=-9\), \(y_2=-2\), \(m = 1\), \(n = 5\).

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

$$
LATEXBLOCK0
$$

Step4: Calculate the y - coordinate of \(B\)

$$
LATEXBLOCK1
$$

So the coordinates of \(B\) are \((1,3)\).

Step5: Identify values for problem 7

For the line - segment \(DF\) with \(D(-1,11)\) and \(F(-9,-5)\) and the ratio \(m:n = 5:8\), \(x_1=-1\), \(y_1 = 11\), \(x_2=-9\), \(y_2=-5\), \(m = 5\), \(n = 8\).

Step6: Calculate the x - coordinate of \(E\)

$$
LATEXBLOCK2
$$

Step7: Calculate the y - coordinate of \(E\)

$$
LATEXBLOCK3
$$

So the coordinates of \(E\) are \((-\frac{53}{13},\frac{63}{13})\).

Answer:

1. \((1,3)\)
2. \((-\frac{53}{13},\frac{63}{13})\)