Question

the coordinates of the endpoints of \\( \overline{tu} \\) are \\( t(-8, -9) \\) and \\( u(1, 9) \\). point \\( v \\) is on \\( \overline{tu} \\) and divides it such that \\( tv:uv \\) is \\( 5:4 \\).
what are the coordinates of \\( v \\)?
write your answers as integers or decimals.
\\( (\square, \square) \\)
submit

Explanation:

Step1: Recall the section formula

The section formula for a point \( V(x, y) \) that divides the line segment joining \( T(x_1, y_1) \) and \( U(x_2, y_2) \) 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, \( T(-8, -9) \), \( U(1, 9) \), \( m = 5 \), \( n = 4 \).

Step2: Calculate the x - coordinate of V

Substitute \( x_1=-8 \), \( x_2 = 1 \), \( m = 5 \), \( n = 4 \) into the formula for \( x \):
\[
x=\frac{5\times1+4\times(-8)}{5 + 4}=\frac{5-32}{9}=\frac{-27}{9}=- 3
\]

Step3: Calculate the y - coordinate of V

Substitute \( y_1=-9 \), \( y_2 = 9 \), \( m = 5 \), \( n = 4 \) into the formula for \( y \):
\[
y=\frac{5\times9+4\times(-9)}{5 + 4}=\frac{45-36}{9}=\frac{9}{9}=1
\]

Answer:

\((-3, 1)\)