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What are the [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-coordinates of point [tex]\( P \)[/tex] on the directed line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] such that [tex]\( P \)[/tex] is [tex]\(\frac{2}{3}\)[/tex] the length of the line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex]?

[tex]\[
\begin{array}{l}
x=\left(\frac{m}{m+n}\right)(x_2-x_1) + x_1 \\
y=\left(\frac{m}{m+n}\right)(y_2-y_1) + y_1
\end{array}
\][/tex]

A. [tex]\((2, -1)\)[/tex]

B. [tex]\((4, -3)\)[/tex]

C. [tex]\((-1, 2)\)[/tex]

D. [tex]\((3, -2)\)[/tex]

Sagot :

GinnyAnsweridn
To find the coordinates of point [tex]\( P \)[/tex] on the directed line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] such that [tex]\( P \)[/tex] is [tex]\(\frac{2}{3}\)[/tex] the length of the line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex], we will use the section formula. Given coordinates:
- Point [tex]\( A \)[/tex] has coordinates [tex]\((2, -1)\)[/tex]
- Point [tex]\( B \)[/tex] has coordinates [tex]\((4, -3)\)[/tex]

The ratio [tex]\( P \)[/tex] divides the line segment [tex]\( AB \)[/tex] is [tex]\( \frac{2}{3} \)[/tex]. We interpret this as [tex]\( P \)[/tex] dividing [tex]\( AB \)[/tex] in the ratio [tex]\( 2:1 \)[/tex] because it is [tex]\(\frac{2}{3}\)[/tex] of the way, meaning [tex]\( \frac{2}{3} \cdot 3 = 2\)[/tex].

### Using the Section Formula
The section formula states:
[tex]\[ \begin{aligned} x_P & = \frac{m \cdot x2 + n \cdot x1}{m + n}, \\ y_P & = \frac{m \cdot y2 + n \cdot y1}{m + n}, \end{aligned} \][/tex]
where [tex]\( A = (x1, y1) \)[/tex], [tex]\( B = (x2, y2) \)[/tex], and [tex]\( m \)[/tex] and [tex]\( n \)[/tex] are the ratios.

In our case:
- [tex]\( x1 = 2 \)[/tex],
- [tex]\( y1 = -1 \)[/tex],
- [tex]\( x2 = 4 \)[/tex],
- [tex]\( y2 = -3 \)[/tex],
- [tex]\( m = 2 \)[/tex],
- [tex]\( n = 1 \)[/tex].

Substituting these values into the formula, we find:
[tex]\[ x_P = \left(\frac{2}{2+1}\right) \left(x2 - x1\right) + x1 \][/tex]
[tex]\[ y_P = \left(\frac{2}{2+1}\right) \left(y2 - y1\right) + y1 \][/tex]

Substituting the coordinates of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ x_P = \left(\frac{2}{2 + 1}\right) \left(4 - 2\right) + 2 = \left(\frac{2}{3}\right) \cdot 2 + 2 = \frac{4}{3} + 2 = \frac{4}{3} + \frac{6}{3} = \frac{10}{3} \approx 3.33 \][/tex]

[tex]\[ y_P = \left(\frac{2}{2 + 1}\right) \left(-3 - (-1)\right) + (-1) = \left(\frac{2}{3}\right) \cdot (-2) + (-1) = \left(\frac{2}{3}\right) \cdot -2 - 1 = -\frac{4}{3} - 1 = -\frac{4}{3} - \frac{3}{3} = -\frac{7}{3} \approx -2.33 \][/tex]

Thus, the coordinates of point [tex]\( P \)[/tex] are approximately:
[tex]\[ (3.333333333333333, -2.333333333333333) \][/tex]

So, the coordinates of point [tex]\( P \)[/tex] are [tex]\( (3, -2) \)[/tex].
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