Question

point d partitions a line segment $overline{bc}$ with endpoints $b(8, 9)$ and $c (16, -7)$ in the ratio such that $bd:dc = 1:7$. what are the coordinates of point d? d (, )

Explanation:

Step1: Use the section - formula for x - coordinate

The formula for the x - coordinate 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 \(x=\frac{mx_2+nx_1}{m + n}\). Here, \(x_1 = 8\), \(x_2=16\), \(m = 1\), and \(n = 7\).

$$x=\frac{1\times16+7\times8}{1 + 7}=\frac{16 + 56}{8}=\frac{72}{8}=9$$

Step2: Use the section - formula for y - coordinate

The formula for the y - coordinate 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 \(y=\frac{my_2+ny_1}{m + n}\). Here, \(y_1 = 9\), \(y_2=-7\), \(m = 1\), and \(n = 7\).

$$y=\frac{1\times(-7)+7\times9}{1 + 7}=\frac{-7 + 63}{8}=\frac{56}{8}=7$$

Answer:

\((9,7)\)