What are the coordinates of the point on the directed line segment from ( 2 , − 6 ) to ( 6 , 2 ) that partitions the segment into a ratio of 3 to 5?

Answer 1

I'm going to use a trick I call "part-to-part-to-whole". It makes ratio problems super-easy. Just remember that the whole is the sum of the parts. In this problem the compound ratio, part:part:whole is 3:5:8. This means that the point we're looking for is 3/8 of the way from (2, -6) and (6, 2)

Since the whole "path" from one point to the other is 4 units to the right and 8 units up, we just need to determine what 3/8 of that is. 3/8 of 4 is 3/2 or 1.5 and 3/8 of 8 is 3. therefore, we add 1.5 to the original x and 3 to the y. The point we need is (3.5, -3)

Answer 2

Think about this as a stick. To find the spot on the stick where the ratio of lengths is 3:5, you could measure the length of the stick and multiple that length by (3/5) to find how far to measure from one side of the stick.

But another way you can think about it is cutting that stick into 8 equal pieces. Put three pieces in one pile and then 5 in another. The ratio of the total length of each pile is 3:5.

So what I did was use the midpoint formula to "slice" the segment into 8 equal pieces.

Let's label point (2,-6) as A and point (6,2) as B. Use the midpoint formula to find point C (4,-2)

Use the midpoint formula to find the midpoint between point A and point C and so on...

Cut the segment in half, then into fourths, then into 8ths.

Does this help?