Question

in the diagram below, $overline{ac}$ has endpoints with coordinates $a(-5,2)$ and $c(4,-10)$. if $b$ is a point on $overline{ac}$ and $ab:bc = 1:2$, what are the coordinates of $b$? step 1: label $(x_1,y_1)$ and $(x_2,y_2)$ on the given coordinates step 2: find the value of $k=a/(a + b)$ step 3: substitute $x$ and $y$ coordinates to find the partitioned coordinates

Explanation:

Step1: Identify the coordinates and ratio values

Let \(A(x_1,y_1)=(-5,2)\) and \(C(x_2,y_2)=(4, - 10)\), and \(a = 1\), \(b=2\).

Step2: Calculate the value of \(k\)

The formula for \(k=\frac{a}{a + b}\), substituting \(a = 1\) and \(b = 2\) gives \(k=\frac{1}{1+2}=\frac{1}{3}\).

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

The formula for the \(x\) - coordinate of the point that divides a line - segment in the ratio \(a:b\) is \(x=k(x_2 - x_1)+x_1\). Substitute \(x_1=-5\), \(x_2 = 4\), and \(k=\frac{1}{3}\):
\[

$$\begin{align*} x&=\frac{1}{3}(4-(-5))+(-5)\\ &=\frac{1}{3}(4 + 5)-5\\ &=\frac{1}{3}\times9-5\\ &=3 - 5\\ &=-2 \end{align*}$$

\]

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

The formula for the \(y\) - coordinate of the point that divides a line - segment in the ratio \(a:b\) is \(y=k(y_2 - y_1)+y_1\). Substitute \(y_1 = 2\), \(y_2=-10\), and \(k=\frac{1}{3}\):
\[

$$\begin{align*} y&=\frac{1}{3}(-10 - 2)+2\\ &=\frac{1}{3}\times(-12)+2\\ &=-4 + 2\\ &=-2 \end{align*}$$

\]

Answer:

\((-2,-2)\)