Partitioning a Segment Quick Check 1 of 51 of 5 Items Question Use the image to answer the question. Calculate point P on AB¯¯¯¯¯¯¯¯ such that the ratio of AP to PB is 1:3. (1 point) Responses P(−212,−214) upper P left parenthesis negative Start Fraction 5 over 2 End Fraction comma negative Start Fraction 9 over 2 End Fraction right parenthesis P(−245,−235) upper P left parenthesis negative Start Fraction 14 over 5 End Fraction comma negative Start Fraction 13 over 5 End Fraction right parenthesis P(12,112) upper P left parenthesis Start Fraction 1 over 2 End Fraction comma Start Fraction 3 over 2 End Fraction right parenthesis P(−1,−1) upper P left parenthesis negative 1 comma negative 1 right parenthesis Skip to navigation page 9 of 9 Formula keypad has been closed. Press Control + Backslash to open it again.Formula keypad has been closed. Press Control + Backslash to open it again.Formula keypad has been closed. Press Control + Backslash to open it again.Formula keypad has been closed. Press Control + Backslash to open it again.Formula keypad has been closed. Press Control + Backslash to open it again.

1 answer

To find point \( P \) on segment \( AB \) such that the ratio of \( AP \) to \( PB \) is 1:3, we need to use the section formula.

Let's denote the coordinates of points \( A \) and \( B \) as:

  • \( A(x_1, y_1) \)
  • \( B(x_2, y_2) \)

According to the section formula, the coordinates of point \( P(x, y) \) that divides the line segment joining \( A \) and \( B \) internally in the ratio \( m:n \) is given by: \[ P\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]

In this case, since \( AP:PB = 1:3 \), we have:

  • \( m = 1 \)
  • \( n = 3 \)

Now we'll rewrite the formula specifically for our values: \[ P\left( \frac{1 \cdot x_2 + 3 \cdot x_1}{1 + 3}, \frac{1 \cdot y_2 + 3 \cdot y_1}{1 + 3} \right) = P\left( \frac{x_2 + 3x_1}{4}, \frac{y_2 + 3y_1}{4} \right) \]

To proceed, we need to know the coordinates of \( A \) and \( B \). However, since the coordinates are not provided in your question, please provide the coordinates for points \( A \) and \( B \) so we can calculate \( P \).