Question

the coordinates of the endpoints of \\(\overline{uv}\\) are \\(u(7, 16)\\) and \\(v(14, 9)\\). point \\(w\\) is on \\(\overline{uv}\\) and divides it such that \\(uw:vw\\) is \\(6:1\\). what are the coordinates of \\(w\\)? write your answers as integers or decimals.

Explanation:

Step1: Recall the section formula

The section formula for a point \( W(x,y) \) that divides the line segment joining \( U(x_1,y_1) \) and \( V(x_2,y_2) \) 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, \( U(7,16) \), \( V(14,9) \), \( m = 6 \), \( n=1 \).

Step2: Calculate the x - coordinate of W

Substitute \( x_1 = 7 \), \( x_2=14 \), \( m = 6 \), \( n = 1 \) into the formula for \( x \):
\[
x=\frac{6\times14+1\times7}{6 + 1}=\frac{84 + 7}{7}=\frac{91}{7}=13
\]

Step3: Calculate the y - coordinate of W

Substitute \( y_1 = 16 \), \( y_2 = 9 \), \( m=6 \), \( n = 1 \) into the formula for \( y \):
\[
y=\frac{6\times9+1\times16}{6 + 1}=\frac{54+16}{7}=\frac{70}{7} = 10
\]

Answer:

\((13, 10)\)