What are the coordinates of the point on the directed line segment from (−1,1) to (8,10) that partitions the segment into a ratio of 2 to 1?

Question

Im confused

I need it fast

Answer 1

Partitions the line segment in the

ratio A to B...

the x-coordinate is : x1 + A(x2-x1)/(A+B)

the y-coordinate is: y1 + A(y2-y1)/(A+B)

A=2 B=1

X = -1 + 2*(8 - -1)/(2+1)

= -1 + 2(9)/3

= -1 + 6

= 5

Y = 1 + 2(10-1)/3

= 1 + 2(9)/3

= 1 + 6

= 7

(5,7) is the point that splits the line

segment into the ratio 2 to 1

CHECK:

distance from (-1,1) to (8,10) is

sqrt( 9^2 + 9^2) = sqrt( 2*81) = 9 rad2

distance from (-1,1) to (5,7) is

sqrt( 6^2 + 6^2) = sqrt(2*36) = 6 rad 2

the proportion is:

6 rad 2 / (9 rad 2) = 6/9 = 2/3

yes it checks

(5,7) is the official answer