Question

A point H on a segment with endpoints B(3, −1) and Z(12, 5) partitions the segment in a 5:1 ratio. Find H. You must show all work to receive credit.

Answer 1

this is a little hard to explain, so please feel free to clarify on any piece of information that is confusing

if the segment BZ is partitioned into a 5:1 ratio, using point H, that means BH must be 5 times the length of HZ

illustrated:



if you consider the whole length as 5x + x = 6x, that means BH must be 5x/6x = 5/6 of the whole length, and HZ must be x/6x = 1/6 of the whole length

Answer 2

this partition also applies to the x and y-coordinates

if we consider the distance between the x-coordinates to be some number (a), then the x-coordinate of H is 5/6 of the way between the x-coordinates of points B and Z

same logic with the y-coordinates

so, we can find the 5/6 point by plotting B and Z, then calculating the 5/6 point

Answer 3

considering only the x-direction first:



the x-coordinates are 12 and 3, so the positive difference is simply 12 - 3 = 9

calculating 5/6 of this gives us 9 * (5/6) = 15/2

so the x-coordinate is 15/2 units long (however, 15/2 is not the x-coordinate b/c we're not starting at 0)
simply add 15/2 to the lower x-coordinate (3) to get the new coordinate

repeat this process with the y-coordinates

report your solution as (x-coordinate, y-coordinate)