Question

the coordinates of the endpoints of $overline{ij}$ are $i(3,5)$ and $j(17,19)$. point $k$ is on $overline{ij}$ and divides it such that $ik:jk$ is $3:4$. what are the coordinates of $k$? write your answers as integers or decimals.

Explanation:

Step1: Use the section - formula for x - coordinate

The formula for the x - coordinate of a point \(K(x,y)\) that divides the line - segment joining \(I(x_1,y_1)\) and \(J(x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\). Here, \(x_1 = 3\), \(x_2=17\), \(m = 3\), and \(n = 4\).

$$x=\frac{3\times17+4\times3}{3 + 4}=\frac{51 + 12}{7}=\frac{63}{7}=9$$

Step2: Use the section - formula for y - coordinate

The formula for the y - coordinate of a point \(K(x,y)\) that divides the line - segment joining \(I(x_1,y_1)\) and \(J(x_2,y_2)\) in the ratio \(m:n\) is \(y=\frac{my_2+ny_1}{m + n}\). Here, \(y_1 = 5\), \(y_2=19\), \(m = 3\), and \(n = 4\).

$$y=\frac{3\times19+4\times5}{3 + 4}=\frac{57+20}{7}=\frac{77}{7}=11$$

Answer:

\((9,11)\)