Points segments

Question

Segment AB has coordinates A(-5,9) and B(7, -7). Points P, Q, and T are collinear points on segment AB with coordinates P(-2,5), Q(1, 1), and T(4, -3). Which of the following line segments would contain the point that partitions line segment AB into a ratio of 3:1?

Answer 1

T(4,-3) partitions AB into a ratio of 3:1. It's 3/4 the way down the line from A.

y decreases from 9 to -7, or -16

x increases from -5 to 7 or 12

slope is -16/12 = -4/3

point slope formula give AB equation

y-9=(-4/3)(x+5)

y=-(4/3)x - 20/3 +27/3

y= -4x/3 +7/3

at the point T(4/-3), plug x=4 and y=-3 into the AB line equation above

-3 = -(4)(4)/3 + 7/3 = -(16-4)/3= -12/3 = -3

it works