Question

Line segment XY is a directed line segment beginning at point X(9, 2) and ending at point Y(-6, 2). Find the point Z on the line segment that partitions the line segment into the segments XZ and ZY at a ratio of 5:3.

Answer 1

Answer:

$(-\frac{3}{8},2)$

Step-by-step explanation:

Point Z divides XY into a 5:3 ratio, so Z is 5/3 of the way from X to Y.  That ratio is k, found by writing the numerator of the ratio (5) over the sum of the numerator and the denominator (5 + 3 = 8).  Our k value is 5/8.  Now we will find the rise and run values which is the slope of this line segment:

$m=\frac{2-2}{-6-9} =\frac{0}{-15}$

Coordinates are found in this formula:

$Z(x,y)=(x_{1}+k(run),y_{1} +k(rise))$

Filling that in:

$Z(x,y)=(9+\frac{5}{8}(-15),2+\frac{5}{8}(0))$

which simplifies to

$Z(x,y)=(9-\frac{75}{8},2+0)$

which gives us the final coordinates of Z to be $Z(x,y)=(-\frac{3}{8},2)$