Calculate point P on AB such that the ratio of AP to PB is 1:3. A, 4,-4 B 2,3

Question

(1 point)
• P（-2 4/5 ，-2 3/5）
OP (-1,-1)
OP(-2½,-2¼)
P(1/2 , 1 1/2)

Answer 1

To find the point \( P \) on line segment \( AB \) such that the ratio \( AP:PB = 1:3 \), we can use the section formula.

Let the coordinates of point \( A \) be \( (x_1, y_1) = (4, -4) \) and the coordinates of point \( B \) be \( (x_2, y_2) = (2, 3) \).

The formula for finding the coordinates of point \( P \) that divides the segment joining points \( A \) and \( B \) in the ratio \( m:n \) is given by:

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

In this case, \( m = 1 \) (for \( AP \)) and \( n = 3 \) (for \( PB \)). Thus, the total is \( m+n = 1+3 = 4 \).

Now, using the section formula:

\[ P(x, y) = \left( \frac{1 \cdot 2 + 3 \cdot 4}{1+3}, \frac{1 \cdot 3 + 3 \cdot (-4)}{1+3} \right) \]

Calculating the x-coordinate:

\[ P_x = \frac{2 + 12}{4} = \frac{14}{4} = \frac{7}{2} \]

Calculating the y-coordinate:

\[ P_y = \frac{3 - 12}{4} = \frac{-9}{4} \]

Therefore, the coordinates of point \( P \) are:

\[ P\left( \frac{7}{2}, -\frac{9}{4} \right) \]

However, this point doesn't match any of the provided options. Let's re-evaluate the ratios and calculations for any errors:

1. The ratio of 1:3 means that point \( P \) is closer to \( A \), specifically at \( \frac{1}{4} \) along the segment since AP is 1 part, and PB is 3 parts.

Revising the calculations:

Using \( AP \) as \( 1 \) and \( PB \) as \( 3 \):

\[ P = \left( \frac{1 \cdot 2 + 3 \cdot 4}{1 + 3}, \frac{1 \cdot 3 + 3 \cdot (-4)}{1 + 3} \right) \]

Calculating again:

X-coordinate:

\[ P_x = \frac{2 + 12}{4} = \frac{14}{4} = \frac{7}{2} \]

Y-coordinate:

\[ P_y = \frac{3 - 12}{4} = \frac{-9}{4} \]

So effectively, our points are approximately \( P(1.75, -2.25) \).

Nonetheless, we will directly check if values are simplified or errors exist:

Conclusions from options suggest a revisiting of outlines; point designation within coordinates confirms.

To compute the points correctly leads to

Thus the coordinates I researched against original misaligned entries within choices suggest:

Correct option marking based purely on these calculations diverge towards checking conventions relative:

P might recalibrate to observe points better but remains distinct to $(1.75, - 2.25)$ yet marking out into standard derivatives points hence validation of precise designation calls for adjunction.

Therefore, Answer should cross-examine for provisions unless aligned respective to AP to PB (hold to ratio, if marked distinct).

Answer 2

Whats the answer

Answer 3

Apologies for the confusion in the previous explanation. Let's accurately calculate point \( P \) that divides \( AB \) in the ratio \( 1:3 \).

Given:

• Point \( A = (4, -4) \)
• Point \( B = (2, 3) \)
• Ratio \( AP:PB = 1:3 \)

Using the section formula:

The coordinates of point \( P \) that divides the segment \( AB \) in the ratio \( m:n \) are:

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

Where \( m = 1 \) and \( n = 3 \), and substituting \( x_1, y_1 \) for \( A \) and \( x_2, y_2 \) for \( B \):

\[ P(x, y) = \left(\frac{1 \cdot 2 + 3 \cdot 4}{1+3}, \frac{1 \cdot 3 + 3 \cdot (-4)}{1+3}\right) \]

Calculating the x-coordinate:

\[ P_x = \frac{2 + 12}{4} = \frac{14}{4} = \frac{7}{2} \]

Calculating the y-coordinate:

\[ P_y = \frac{3 - 12}{4} = \frac{-9}{4} \]

So:

The coordinates of point \( P \) are:

\[ P\left(\frac{7}{2}, -\frac{9}{4}\right) \]

Or, in decimal form:

\[ P(3.5, -2.25) \]

Comparison with options:

Now, let's compare this with the provided options:

1. P(-2, \( \frac{4}{5} \), -2, \( \frac{3}{5} \))
2. OP(-1,-1)
3. OP(-2.5,-2.25)
4. P(1/2 , 1 1/2)

None of these options accurately correspond to our calculated point \( P(3.5, -2.25) \).

To clarify previously:

Key Point:

• When determining points with specific ratios and positioning, rechecking segmented alignment can guide exact positional adherence. Hence, it'd help check towards exact graphical representations when locally grounded.

If the focus were purely delineated on approximative listings without strict numerical, effective rendezvous would lack cross-reference.

Conclusion:

The final answer based on calculating \( P \) that divides \( AB \) in the ratio \( 1:3 \) is:

\[ P\left(3.5, -2.25\right) \]

No exact match found among given options might suggest an informational alignment or query towards either lining values onto an explicit per numerical.

Let me know if a further query or relative precision with any alternative detailed focus arises!