Question

Determine the coordinates of point  P  on  AB¯¯¯¯¯¯¯¯  such that the ratio of  AP  to  PB  is 1:3.(1 point)P is located at (,).
points :(0,-1) (0,-5)

Answer 1

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

Given points \( A(0, -1) \) and \( B(0, -5) \), we can denote the coordinates of point \( P(x, y) \).

Since both points \( A \) and \( B \) have the same x-coordinate of 0, we are dealing with a vertical line segment along the y-axis.

The section formula states that if a point \( P \) divides the line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \), the coordinates of point \( P \) can be found as follows:

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

Here, \( A(0, -1) \) and \( B(0, -5) \), and we want \( AP:PB = 1:3\) so \( m = 1 \) and \( n = 3\).

Using the section formula, we calculate the y-coordinate:

\[
P_y = \frac{1 \cdot (-5) + 3 \cdot (-1)}{1 + 3} = \frac{-5 - 3}{4} = \frac{-8}{4} = -2
\]

Since the x-coordinates of both points are zero, the x-coordinate of point \( P \) will also be 0:

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

\[
P(0, -2)
\]

Thus, point \( P \) is located at \((0, -2)\).