The problem asks to find the coordinates of point $P$ that divides the line segment $AB$ internally in the ratio 2:1, where:

• Point $A(3, 7)$
• Point $B(4, 8)$

Formula for the Section Formula:

If a point $P(x, y)$ divides a line segment joining points $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $m:n$, the coordinates of $P$ are given by:

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

Here, $m:n = 2:1$, $A(3, 7)$, and $B(4, 8)$.

Substituting values:

$x = \frac{2(4) + 1(3)}{2+1} = \frac{8 + 3}{3} = \frac{11}{3} \approx 3.67$ $y = \frac{2(8) + 1(7)}{2+1} = \frac{16 + 7}{3} = \frac{23}{3} \approx 7.67$

Thus, the coordinates of point $P$ are approximately $P(3.67, 7.67)$.

Would you like a more detailed explanation, or do you have any questions?

Here are some related questions to explore:

1. How would the formula change if the ratio were 3:2?
2. What would happen if the point divided the segment externally?
3. How do you calculate the midpoint of a line segment?
4. Can the section formula be used in three dimensions?
5. How is the section formula derived from the concept of weighted averages?

Tip: The section formula is helpful in many geometric problems involving line segments and ratios. Understanding it can be useful for higher-dimensional problems as well!