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

A. [tex]\((2, -1)\)[/tex]
B. [tex]\((4, -3)\)[/tex]
C. [tex]\((-1, 2)\)[/tex]
D. [tex]\((3, -2)\)[/tex]

Sagot :

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] of the length of the segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex], we can use the section formula for internal division.

Given points [tex]\( A(x_1, y_1) = (2, -1) \)[/tex] and [tex]\( B(x_2, y_2) = (4, -3) \)[/tex], let's calculate the coordinates step-by-step:

1. We know that [tex]\( P \)[/tex] divides the segment [tex]\( AB \)[/tex] in the ratio [tex]\( \frac{2}{3} \)[/tex]. Therefore, we can denote [tex]\( m = 2 \)[/tex] and [tex]\( n = 1 \)[/tex]. This comes from the ratio [tex]\( 2 \)[/tex] parts of [tex]\( AB \)[/tex] to [tex]\( 1 \)[/tex] part of [tex]\( AP \)[/tex] added together giving [tex]\( m + n = 3 \)[/tex].

2. Substitute the ratio and coordinates into the section formula:
[tex]\[ x = \left(\frac{m}{m+n}\right)(x_2 - x_1) + x_1 \][/tex]
[tex]\[ y = \left(\frac{m}{m+n}\right)(y_2 - y_1) + y_1 \][/tex]

Plugging in the values:
[tex]\[ x = \left(\frac{2}{3}\right)(4 - 2) + 2 \][/tex]

Now calculate the [tex]\( x \)[/tex]-coordinate:
[tex]\[ x = \left(\frac{2}{3}\right)(2) + 2 = \frac{4}{3} + 2 = \frac{4}{3} + \frac{6}{3} = \frac{10}{3} \approx 3.33 \][/tex]

Now for the [tex]\( y \)[/tex]-coordinate:
[tex]\[ y = \left(\frac{2}{3}\right)(-3 - (-1)) + (-1) \][/tex]
[tex]\[ y = \left(\frac{2}{3}\right)(-2) - 1 = -\frac{4}{3} - 1 = -\frac{4}{3} - \frac{3}{3} = -\frac{7}{3} \approx -2.33 \][/tex]

Therefore, the coordinates of point [tex]\( P \)[/tex] are:
[tex]\[ \left( 3.33, -2.33 \right) \][/tex]