Question

Find the coordinates of point X if it partitions the directed segment AB¯¯¯¯¯¯¯¯ in the ratio 1:2 . (1 point) Responses X(−213,−113) upper X left parenthesis negative Start Fraction 7 over 3 End Fraction comma negative 2 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(−1,−23) upper X left parenthesis negative 1 comma negative Start Fraction 2 over 3 End Fraction right parenthesis X(−5,−423)

Answer 1

A is located at point (-4, -3) and B is located at point (1, 2)

Answer 2

The options given are,
A: X(-2 1/3, -1 1/3)
B: X(-5 1/4, -2 1/2)
C: X(-1, -2/3)
D: X(-5, -4 2/3)

Answer 3

To find the coordinates of point \( X \) that partitions the directed segment \( \overline{AB} \) in the ratio \( 1:2 \), we can use the section formula.

Let the coordinates of points \( A \) and \( B \) be given as \( A(x_1, y_1) \) and \( B(x_2, y_2) \). The formula for dividing a segment in the ratio \( m:n \) is given by:

\[
X = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\]

In this case, we have \( m = 1 \) and \( n = 2 \).

Let's denote:
- \( A(x_1, y_1) \)
- \( B(x_2, y_2) \)

We can find the coordinates of \( X \) using the formula:

\[
X = \left( \frac{1 \cdot x_2 + 2 \cdot x_1}{1+2}, \frac{1 \cdot y_2 + 2 \cdot y_1}{1+2} \right) = \left( \frac{x_2 + 2x_1}{3}, \frac{y_2 + 2y_1}{3} \right)
\]

To determine \( X \) specifically, we need the coordinates of points \( A \) and \( B \). However, since those coordinates are not provided, we will evaluate each of the answer choices as potential candidates for coordinates of \( A \) and \( B \).

1. \( X(-213, -113) \)
2. \( X\left(-\frac{7}{3}, -2\right) \)
3. \( X(-514, -212) \)
4. \( X\left(-\frac{21}{4}, -\frac{5}{2}\right) \)
5. \( X(-1, -\frac{2}{3}) \)
6. \( X(-5, -423) \)

We need more information on points \( A \) and \( B \) to select any of the above as a correct answer unless specific coordinates are given for \( A \) and \( B \).

Please provide the coordinates of points \( A \) and \( B \) so that we can find the correct coordinates for \( X \). If that's not available, then I can't directly solve for the coordinates of \( X \).

Answer 4

Given points \( A(-4, -3) \) and \( B(1, 2) \), we want to find the coordinates of point \( X \) that partitions the directed segment \( \overline{AB} \) in the ratio \( 1:2 \).

Using the section formula:

\[
X = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\]

Here, \( m = 1 \), \( n = 2 \), and the coordinates for \( A \) and \( B \) are:
- \( A(x_1, y_1) = (-4, -3) \)
- \( B(x_2, y_2) = (1, 2) \)

Now substituting the values into the formula:

### For the x-coordinate:
\[
X_x = \frac{1 \cdot 1 + 2 \cdot (-4)}{1 + 2} = \frac{1 - 8}{3} = \frac{-7}{3}
\]

### For the y-coordinate:
\[
X_y = \frac{1 \cdot 2 + 2 \cdot (-3)}{1 + 2} = \frac{2 - 6}{3} = \frac{-4}{3}
\]

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

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

Now, let's match this with the provided options. The value \( -\frac{7}{3} \) corresponds to the x-coordinate, and \( -\frac{4}{3} \) corresponds to the y-coordinate but is not listed among the options. However, there seems to be a small mistake since the expected coordinate for y was \( -2 \) rather than \( -\frac{4}{3} \).

To conclude, we computed correctly, and \( X \) is indeed at:

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

It seems that none of the provided options exactly match this correct answer. Please verify the options given.

Answer 5

Thanks for providing the correct options for point \( X \). Let's clarify the partitioning and check the calculations again to find the correct coordinates.

Given points \( A(-4, -3) \) and \( B(1, 2) \) partitioned in the ratio \( 1:2 \):

Using the section formula:

\[
X = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\]

Where:
- \( m = 1 \) (the part towards \( B \)),
- \( n = 2 \) (the part towards \( A \)),
- \( A(-4, -3) \) and \( B(1, 2) \).

We can calculate the x-coordinate:

### X-coordinate
\[
X_x = \frac{1 \cdot 1 + 2 \cdot (-4)}{1 + 2} = \frac{1 - 8}{3} = \frac{-7}{3} \approx -2.33
\]

### Y-coordinate
\[
X_y = \frac{1 \cdot 2 + 2 \cdot (-3)}{1 + 2} = \frac{2 - 6}{3} = \frac{-4}{3} \approx -1.33
\]

So we have:

\[
X\left(-\frac{7}{3}, -\frac{4}{3}\right) \approx X(-2.33, -1.33)
\]

Now, let's match this with the options you provided:

1. **A:** \( X(-2 \frac{1}{3}, -1 \frac{1}{3}) = X(-2.33, -1.33) \)
2. **B:** \( X(-5 \frac{1}{4}, -2 \frac{1}{2}) \)
3. **C:** \( X(-1, -\frac{2}{3}) \)
4. **D:** \( X(-5, -4 \frac{2}{3}) \)

The closest match is:

**A:** \( X(-2 \frac{1}{3}, -1 \frac{1}{3})\)

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

\[
X(-2 \frac{1}{3}, -1 \frac{1}{3})
\]

This matches our earlier calculated \( X(-2.33, -1.33) \). Therefore, the correct answer is **A**.