Question

Find the coordinates of point X if it partitions the directed segment AB¯¯¯¯¯¯¯¯ in the ratio 1:2 . (1 point) Responses X(−1,−23) upper X left parenthesis negative 1 comma negative Start Fraction 2 over 3 End Fraction right parenthesis X(−514,−212) upper X left parenthesis negative Start Fraction 21 over 4 End Fraction comma negative Start Fraction 5 over 2 End Fraction right parenthesis X(−5,−423) upper X left parenthesis negative 5 comma negative Start Fraction 14 over 3 End Fraction right parenthesis X(−213,−113)
A (-4, -3)
B(1,2)
Select One

Answer 1

To find the coordinates of point \( X \) that partitions the directed segment \( AB \) in the ratio \( 1:2 \), we can use the formula for the coordinates of a point that divides a segment in a ratio \( m:n \):

\[
X = \left( \frac{n \cdot x_1 + m \cdot x_2}{m+n}, \frac{n \cdot y_1 + m \cdot y_2}{m+n} \right)
\]

where \( A(x_1, y_1) \) and \( B(x_2, y_2) \) are the coordinates of points \( A \) and \( B \), respectively, and \( m \) and \( n \) are the parts of the ratio.

Given:
- \( A(-4, -3) \)
- \( B(1, 2) \)
- \( m = 1 \)
- \( n = 2 \)

We can plug in the coordinates into the formula:

1. Calculate the x-coordinate of \( X \):

\[
x_X = \frac{n \cdot x_1 + m \cdot x_2}{m+n} = \frac{2 \cdot (-4) + 1 \cdot 1}{1 + 2} = \frac{-8 + 1}{3} = \frac{-7}{3}
\]

2. Calculate the y-coordinate of \( X \):

\[
y_X = \frac{n \cdot y_1 + m \cdot y_2}{m+n} = \frac{2 \cdot (-3) + 1 \cdot 2}{1 + 2} = \frac{-6 + 2}{3} = \frac{-4}{3}
\]

Thus, the coordinates of point \( X \) are:

\[
X \left( \frac{-7}{3}, \frac{-4}{3} \right)
\]

Now we can match this against the answer choices provided:
- \( X\left( -1, -23 \right) \)
- \( X\left( -1, -\frac{2}{3} \right) \)
- \( X\left( -514, -212 \right) \)
- \( X\left( -\frac{21}{4}, -\frac{5}{2} \right) \)
- \( X\left( -5, -\frac{14}{3} \right) \)
- \( X\left( -213, -113 \right) \)

None of the answers match \( X \left( -\frac{7}{3}, -\frac{4}{3} \right) \). It appears there may be a mistake in the choices provided, as our calculations hold true based on the specified ratio dividing the segment.

If we double-check the results, our calculations conclude with \( X\left( -\frac{7}{3}, -\frac{4}{3} \right) \), which does not appear to be an option provided for selection.

If there's any further instruction or detail regarding this task or options available, please share!