Point B is at (3,1). For which segment is B a midpoint?

AC¯¯¯¯¯¯¯¯ with endpoints at A(1,−1) and C(8,6)
AC¯¯¯¯¯¯¯¯ with endpoints at A(1,−1) and C(5,3)
AD¯¯¯¯¯¯¯¯ with endpoints at A(1,−1) and C(6,4)
AC¯¯¯¯¯¯¯¯ with endpoints at A(1,−1) and C(7,5)

To determine which segment Point B is a midpoint of, we can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) can be found using the following equations:

Midpoint x-coordinate = (x₁ + x₂) / 2
Midpoint y-coordinate = (y₁ + y₂) / 2

Let's calculate the midpoints for each given line segment:

AC¯¯¯¯¯¯¯¯ with endpoints at A(1, -1) and C(8, 6):
Midpoint x-coordinate = (1 + 8) / 2 = 9 / 2 = 4.5
Midpoint y-coordinate = (-1 + 6) / 2 = 5 / 2 = 2.5

AC¯¯¯¯¯¯¯¯ with endpoints at A(1, -1) and C(5, 3):
Midpoint x-coordinate = (1 + 5) / 2 = 6 / 2 = 3
Midpoint y-coordinate = (-1 + 3) / 2 = 2 / 2 = 1

AD¯¯¯¯¯¯¯¯ with endpoints at A(1, -1) and D(6, 4):
Midpoint x-coordinate = (1 + 6) / 2 = 7 / 2 = 3.5
Midpoint y-coordinate = (-1 + 4) / 2 = 3 / 2 = 1.5

AC¯¯¯¯¯¯¯¯ with endpoints at A(1, -1) and C(7, 5):
Midpoint x-coordinate = (1 + 7) / 2 = 8 / 2 = 4
Midpoint y-coordinate = (-1 + 5) / 2 = 4 / 2 = 2

Comparing the coordinates of Point B (3, 1) with the calculated midpoints, we see that Point B is a midpoint of the segment AC¯¯¯¯¯¯¯¯ with endpoints at A(1, -1) and C(5, 3).

well, you could test each option and find their midpoints

or
you could let the other endpoint be (x,y)(3,1)
then ( (x+1)/2 , (y-1)/2) = (3,1)

(x+1)/2 = 3 ---> x = 5
(y-1)/2 = 1 ----> y = 3

thank you!

A(1. -1), B(x, y), C(5, 3).

x-1 = 1/2(5-1)
X = 3.

y+1 = 1/2(3+1)
Y = 1.