Question

the coordinates of the endpoints of $overline{bc}$ are $b(-11, 6)$ and $c(5, -10)$. point $d$ is on $overline{bc}$ and divides it such that $bd:cd$ is $7:1$. what are the coordinates of $d$? write your answers as integers or decimals.

Explanation:

Step1: Recall the section - formula

The formula to find the coordinates of a point \(D(x,y)\) that divides the line - segment joining \(B(x_1,y_1)\) and \(C(x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(x_1=-11,y_1 = 6,x_2 = 5,y_2=-10,m = 7,n = 1\).

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

\[

$$\begin{align*} x&=\frac{7\times5+1\times(-11)}{7 + 1}\\ &=\frac{35-11}{8}\\ &=\frac{24}{8}\\ &=3 \end{align*}$$

\]

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

\[

$$\begin{align*} y&=\frac{7\times(-10)+1\times6}{7 + 1}\\ &=\frac{-70 + 6}{8}\\ &=\frac{-64}{8}\\ &=-8 \end{align*}$$

\]

Answer:

\((3,-8)\)