Question

1. directed line segment bn has endpoints at point b(9, -7) and point n(-1, -2). directed line segment bn is partitioned by point h at a 4:1 ratio. what are the coordinates of point h?

Explanation:

Step1: Recall the section - formula

The formula to find the coordinates of a point \(H(x,y)\) that divides the line - segment joining \(B(x_1,y_1)\) and \(N(x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(x_1 = 9,y_1=-7,x_2=-1,y_2 = - 2,m = 4,n = 1\).

Step2: Calculate the \(x\) - coordinate of point \(H\)

\[

$$\begin{align*} x&=\frac{4\times(-1)+1\times9}{4 + 1}\\ &=\frac{-4 + 9}{5}\\ &=\frac{5}{5}\\ &=1 \end{align*}$$

\]

Step3: Calculate the \(y\) - coordinate of point \(H\)

\[

$$\begin{align*} y&=\frac{4\times(-2)+1\times(-7)}{4 + 1}\\ &=\frac{-8-7}{5}\\ &=\frac{-15}{5}\\ &=-3 \end{align*}$$

\]

Answer:

\((1,-3)\)