Question

the coordinates of the endpoints of \\(\overline{lm}\\) are \\(l(5, 17)\\) and \\(m(15, 12)\\). point \\(n\\) is on \\(\overline{lm}\\) and divides it such that \\(ln:mn\\) is \\(2:3\\). what are the coordinates of \\(n\\)? write your answers as integers or decimals. (\\(\square\\), \\(\square\\)) submit

Explanation:

Step1: Recall the section formula

To find the coordinates of a point \( N(x,y) \) that divides the line segment joining \( L(x_1,y_1) \) and \( M(x_2,y_2) \) in the ratio \( m:n \), the section formula is \( x=\frac{mx_2 + nx_1}{m + n} \) and \( y=\frac{my_2+ny_1}{m + n} \). Here, \( L(5,17) \), \( M(15,12) \), \( m = 2 \), \( n=3 \).

Step2: Calculate the x - coordinate of N

Substitute \( x_1 = 5 \), \( x_2=15 \), \( m = 2 \), \( n = 3 \) into the formula for \( x \):
\( x=\frac{2\times15+3\times5}{2 + 3}=\frac{30 + 15}{5}=\frac{45}{5}=9 \)

Step3: Calculate the y - coordinate of N

Substitute \( y_1 = 17 \), \( y_2 = 12 \), \( m = 2 \), \( n = 3 \) into the formula for \( y \):
\( y=\frac{2\times12+3\times17}{2+3}=\frac{24 + 51}{5}=\frac{75}{5}=15 \)

Answer:

\((9, 15)\)