on the directed segment (1,8) to (5,0) find the point on the segment that divides into a ratio of 3:1

Answer 1

Distance between points is:

D = √(5-1)² + (0-8)²) = √(16 + 64) = √(80) = 4√5

Dividing into 2 parts of ratio 3:1 yields part lengths of 3√5 and √5

Answer 2

you want the coordinates of the point that divides the segment into a ratio of 3:1 from point (1,8) to point (5,0)

call this point (xp,yp) and call the ratio a:b

x1=1, y1=8 and x2=5, y2=0

xp=x1+(a/(a+b))(x2-x1)

yp=y1+(a/(a+b))(y2-y1)

xp=1+(3/4)(5-1)=1+(3/4)(4)=4

yp=8+(3/4)(0-8)=8+(3/4)(-8)=2

the point is xp=(4,2)

another solution is to draw a diagram on graph paper

go from B(5,0) to A(1,8)

from B to A you want the coordinates of the point 1/4 of the way up the diagonal line

1/4 of the way from 0 to 8 is 2, which is the y value of the point

1/4 of the way from 5 to 1 is (1/4)*4=1, so you go from 5 to 4 which is a difference of 1, so the x value of the point is 4

again, the point you want is (4,2)