What are the coordinates of the point on the directed line segment from ( 3 , 4 ) (3,4) to ( 9 , 1 ) (9,1) that partitions the segment into a ratio of 2 to 1?

Question

What are the coordinates of the point on the directed line segment from $(3,4)$ to $(9,1)$ that partitions the segment into a ratio of 2 to 1?

Answer 1

To find the point, we need to find the x and y coordinates separately by considering the horizontal and vertical distances between (3,4) and (9,1). The difference between the x coords is 9-3=6. ⅔ of this distance is 4; the difference between the y coords is 4-1=3 and ⅔ of this distance is 2. The line segment slopes downward from left to right, (3,4) to (9,1), so the x coord is going to be 4 units to the right of (3,4) which means x=3+4=7. The y coord is going to be 2 units directly below (3,4) which means y=4-2=2. Put these together and we get (7,2), which is ⅔ along the line segment, dividing the segment from its starting point at (3,4) into the ratio 2:1. The line segment represent the hypotenuse of a right triangle with its right angle at (3,1), and by the rules of similar triangles, if the vertical and horizontal legs are divided into the ratio 2:1, so will the hypotenuse.