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. \\((\quad, \quad)\\) submit

Explanation:

Step1: Recall the section formula

The section formula for a point \( K(x, y) \) dividing the line segment joining \( I(x_1, y_1) \) and \( J(x_2, y_2) \) in the ratio \( m:n \) is given by:
\[
x=\frac{mx_2 + nx_1}{m + n}, \quad y=\frac{my_2 + ny_1}{m + n}
\]
Here, \( I(3, 5) \), \( J(17, 19) \), \( m = 3 \), \( n = 4 \).

Step2: Calculate the x - coordinate of K

Substitute \( x_1 = 3 \), \( x_2 = 17 \), \( m = 3 \), \( n = 4 \) into the formula for \( x \):
\[
x=\frac{3\times17+4\times3}{3 + 4}=\frac{51 + 12}{7}=\frac{63}{7}=9
\]

Step3: Calculate the y - coordinate of K

Substitute \( y_1 = 5 \), \( y_2 = 19 \), \( m = 3 \), \( n = 4 \) into the formula for \( y \):
\[
y=\frac{3\times19+4\times5}{3 + 4}=\frac{57+20}{7}=\frac{77}{7} = 11
\]

Answer:

\((9, 11)\)