To solve for point $P$ on the line segment $\overline{AB}$ such that the ratio of $AP$ to $PB$ is $1:3$, follow these steps:

Step 1: Determine Coordinates of Points $A$ and $B$

From the graph:

• $A$ is at $(-3, -4)$.
• $B$ is at $(2, 3)$.

Step 2: Use the Section Formula

The section formula finds a point dividing a line segment into a given ratio: $P(x, y) = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)$ where:

• $(x_1, y_1)$ are coordinates of $A$,
• $(x_2, y_2)$ are coordinates of $B$,
• $m:n$ is the given ratio.

Step 3: Apply the Ratio $1:3$

Let $m = 1$ and $n = 3$. Thus, $P$ divides $\overline{AB}$ in the ratio $1:3$.

Step 4: Substitute the Values

Substitute into the formula: $x = \frac{(1)(2) + (3)(-3)}{1+3} = \frac{2 - 9}{4} = \frac{-7}{4} = -1.75$ $y = \frac{(1)(3) + (3)(-4)}{1+3} = \frac{3 - 12}{4} = \frac{-9}{4} = -2.25$

Step 5: Final Answer

The coordinates of point $P$ are: $P = \left(-1.75, -2.25\right)$

Would you like further details or have any questions?

Related Questions

1. How can you find the coordinates of a point that divides a segment in any given ratio?
2. What is the formula for finding the midpoint of a line segment?
3. How do you derive the section formula?
4. What changes in the formula if the ratio is reversed (e.g., 3:1)?
5. How do you find the length of a line segment between two points?

Tip

Always verify the coordinates by checking the ratio visually or using a tool, especially when working with graphical representations!