Question

the point b lies on the segment $overline{ac}$. find the coordinates of b so that ab is $\frac{1}{5}$ of ac. a (-22,25) c (3,-5) coordinates of b: ( , )

Explanation:

Step1: Use section - formula

The section - formula for a point \(B(x,y)\) that divides the line - segment joining \(A(x_1,y_1)\) and \(C(x_2,y_2)\) 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, \(m = 1\), \(n=4\), \(x_1=-22\), \(y_1 = 25\), \(x_2 = 3\), and \(y_2=-5\).

Step2: Calculate the x - coordinate of B

\[

$$\begin{align*} x&=\frac{1\times3+4\times(-22)}{1 + 4}\\ &=\frac{3-88}{5}\\ &=\frac{-85}{5}\\ &=-17 \end{align*}$$

\]

Step3: Calculate the y - coordinate of B

\[

$$\begin{align*} y&=\frac{1\times(-5)+4\times25}{1 + 4}\\ &=\frac{-5 + 100}{5}\\ &=\frac{95}{5}\\ &=19 \end{align*}$$

\]

Answer:

\((-17,19)\)