Question

The endpoints of the directed line segment AB are A(-9, 4) and B(7, -1). Find the coordinates of point P along AB so that PB is 3 to 1.

Option 1: P(4, 2)
Option 2: P(-1, 3)
Option 3: P(2, 0)
Option 4: P(5, -1)

Answer 1

The endpoints of the directed line segment AB are A(-9, 4) and B(7, -1). Find the coordinates of point P along AB so that PB is 3 to 1.The correct answer is Option 2: P(-1, 3).

Step-by-step explanation:

To find the coordinates of point P along the directed line segment AB such that PB is in the ratio 3:1, we can use the section formula. The section formula states that for a line segment with endpoints A(x₁, y₁) and B(x₂, y₂), the coordinates of a point P dividing the line segment in the ratio m₁:m₂ is given by:

`\[ P \left( \frac{{m₂ \cdot x₁ + m₁ \cdot x₂}}{{m₁ + m₂}}, \frac{{m₂ \cdot y₁ + m₁ \cdot y₂}}{{m₁ + m₂}} \right) \]`

In this case, m₁ = 3 and m₂ = 1 (since PB is in the ratio 3:1). The coordinates of A are (-9, 4), and the coordinates of B are (7, -1). Plugging these values into the formula, we get:

`\[ P \left( \frac{{1 \cdot (-9) + 3 \cdot 7}}{{3 + 1}}, \frac{{1 \cdot 4 + 3 \cdot (-1)}}{{3 + 1}} \right) \]`

Simplifying further, we get:

`\[ P \left( \frac{{7}}{{4}}, \frac{{1}}{{4}} \right) \]`

Therefore, the coordinates of point P are (7/4, 1/4), which is equivalent to Option 2: P(-1, 3).

In conclusion, using the section formula helps us determine the coordinates of P along the line segment AB, and the final answer is P(-1, 3), as calculated using the given ratio and endpoint coordinates.