To solve this problem, we can use the section formula. The section formula helps find the coordinates of a point that divides a line segment in a given ratio.

The coordinates of the endpoints of $\overline{TU}$ are:

• $T(-2, 7)$
• $U(4, -8)$

The ratio $TV : UV$ is given as $1 : 2$, meaning point $V$ divides the line segment $TU$ in the ratio $m:n = 1:2$.

The section formula is:

$(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)$

Here:

• $x_1 = -2$, $y_1 = 7$ (coordinates of $T$)
• $x_2 = 4$, $y_2 = -8$ (coordinates of $U$)
• $m = 1$, $n = 2$

Substituting these values into the formula:

$x = \frac{1 \cdot 4 + 2 \cdot (-2)}{1+2} = \frac{4 - 4}{3} = 0$ $y = \frac{1 \cdot (-8) + 2 \cdot 7}{1+2} = \frac{-8 + 14}{3} = \frac{6}{3} = 2$

Thus, the coordinates of $V$ are $(0, 2)$.

Would you like further details on the steps, or do you have any questions?

Here are 5 related questions to expand your understanding:

1. How would the coordinates of $V$ change if the ratio were $2:1$?
2. What if the point $V$ divides $TU$ externally instead of internally?
3. Can you apply the section formula in three-dimensional space?
4. How would you find the length of the segment $TV$?
5. How does the midpoint formula relate to the section formula?

Tip: The section formula is highly versatile and works for both internal and external divisions, as long as you adjust the sign for external division.