To determine the coordinates of the point that partitions the line segment from $A(-2, -4)$ to $B(5, 3)$ in the ratio $5:2$, we use the section formula:

$\text{Coordinates of the point } P(x, y) = \left( \frac{m_2x_1 + m_1x_2}{m_1 + m_2}, \frac{m_2y_1 + m_1y_2}{m_1 + m_2} \right)$

Where:

• $A(x_1, y_1) = (-2, -4)$
• $B(x_2, y_2) = (5, 3)$
• $m_1 : m_2 = 5:2$

Step 1: Compute the $x$-coordinate

$x = \frac{m_2x_1 + m_1x_2}{m_1 + m_2} = \frac{2(-2) + 5(5)}{5 + 2} = \frac{-4 + 25}{7} = \frac{21}{7} = 3$

Step 2: Compute the $y$-coordinate

$y = \frac{m_2y_1 + m_1y_2}{m_1 + m_2} = \frac{2(-4) + 5(3)}{5 + 2} = \frac{-8 + 15}{7} = \frac{7}{7} = 1$

Thus, the coordinates of the point are: $P(3, 1)$

Step 3: Horizontal component

The horizontal component is the $x$-coordinate of the point, which is: $x = 3$

Final Answer:

1. The point that partitions the segment in a ratio of $5:2$ is $P(3, 1)$.
2. The horizontal component is $3$.

Would you like a visual explanation or further clarification? 😊

---

Related Questions:

1. How is the section formula derived?
2. What happens if the ratio is reversed to $2:5$?
3. How do we calculate the distance of the partitioned point from $A$ or $B$?
4. What is the midpoint of the line segment $AB$?
5. Can the section formula be applied in 3D geometry?

Tip:

The section formula works not only for directed line segments but also for weights in physics, where the ratio represents proportional forces or moments.