### Video Transcript

If the coordinates of points 𝐴 and
𝐵 are negative two, negative two and two, negative four, respectively, and point 𝑃
lies on the line segment 𝐴𝐵 such that 𝐴𝑃 is equal to three-sevenths of 𝐴𝐵,
find the coordinates of 𝑃.

Let’s first consider the line 𝐴𝐵,
where 𝐴 has coordinates negative two, negative two and 𝐵 has coordinates two,
negative four. We’re told that 𝐴𝑃 is
three-sevenths of the length of 𝐴𝐵. Therefore, point 𝑃 lies
three-sevenths of the way along the line. This means that length 𝑃𝐵 must be
four-sevenths of the length 𝐴𝑃. This means that the ratio of 𝐴𝑃
to 𝑃𝐵 must be three to four.

We know that if point 𝑃 lies on
the line segment 𝐴𝐵, where 𝐴𝑃 and 𝑃𝐵 have ratio 𝑚 to 𝑛, then the coordinates
of point 𝑃 will be given by the following. The 𝑥-coordinate will be 𝑛
multiplied by 𝑥 one plus 𝑚 multiplied by 𝑥 two divided by 𝑚 plus 𝑛. The 𝑦-coordinate will be given by
𝑛 multiplied by 𝑦 one plus 𝑚 multiplied by 𝑦 two divided by 𝑚 plus 𝑛, where
point 𝐴 has coordinates 𝑥 one, 𝑦 one and point 𝐵 has coordinates 𝑥 two, 𝑦
two. In our case, 𝑥 one, 𝑦 one is
negative two, negative two and 𝑥 two, 𝑦 two is two, negative four. The ratios 𝑚 and 𝑛 are three and
four, respectively.

The 𝑥-coordinate of point 𝑃 is
therefore equal to four multiplied by negative two plus three multiplied by two all
divided by three plus four. Four multiplied by negative two is
negative eight. Three multiplied by two is equal to
six, and three plus four is equal to seven. We’re therefore left with negative
eight plus six divided by seven. This is equal to negative
two-sevenths. The 𝑥-coordinate of 𝑃 is negative
two-sevenths.

The 𝑦-coordinate can be calculated
in the same way: four multiplied by negative two plus three multiplied by negative
four divided by three plus four. Four multiplied by negative two is
negative eight, and three multiplied by negative four is negative 12. Three plus four is equal to
seven. Negative eight minus 12 divided by
seven is equal to negative twenty sevenths. Therefore, the 𝑦-coordinate of 𝑃
is negative twenty sevenths.

The coordinates of the point 𝑃
that lies on the line segment 𝐴𝐵 are negative two-sevenths, negative twenty
sevenths.