To find the coordinates of point [tex]\( P \)[/tex] that partitions the line segment [tex]\( AB \)[/tex] into a part-to-whole ratio of [tex]\( 1:5 \)[/tex], we first need to understand what this ratio represents and how to use it in our calculations.
Given:
- Line segment [tex]\( AB \)[/tex] with endpoints at [tex]\( A(-9, 3) \)[/tex] and [tex]\( B(1, 8) \)[/tex].
- Part-to-whole ratio [tex]\( 1 : 5 \)[/tex].
In order to determine the correct placement of point [tex]\( P \)[/tex] along segment [tex]\( AB \)[/tex], we need to convert this part-to-whole ratio into a fraction.
The ratio [tex]\( 1 : 5 \)[/tex] means that if we were to divide the segment into a total of 6 equal parts (since [tex]\( 1 + 5 = 6 \)[/tex]), point [tex]\( P \)[/tex] would be placed after the first part when moving from [tex]\( A \)[/tex] to [tex]\( B \)[/tex].
This translates to the fraction of the distance from [tex]\( A \)[/tex] to [tex]\( P \)[/tex] relative to the entire segment [tex]\( AB \)[/tex]. The fraction is:
[tex]\[ \frac{m}{m+n} = \frac{1}{1+5} = \frac{1}{6} \][/tex]
Therefore, the fraction [tex]\(\frac{m}{m+n} \)[/tex] equals [tex]\( \frac{1}{6} \)[/tex].
