Two birds are flying toward a birdhouse that is halfway between them. The birds are at coordinates A (-6, 6) and B (8,-4). What are the coordinates of the birdhouse?

Question

Given that AB ∦ DC (not parallel)

A(8,3), B(6,7)

C(6,0), D(-5,1)

Answer 1

the half way position is the midpoint between (-6,6) and (8,-4) calculated by taking the average of the x coordinates (-6+8)/2 = 1

and the average of the y coordinates (6-4)/2 = 1.

the midpoint is (1,1) but that's not one of the 4 choices, A, B, C or D.

C's (6,0) is closet to (1,1)

a graph of the line connecting the positions of the 2 birds is (y-6)/(x+6) = (6+4)/(-6-8) = -5/7

or 7y-42 = -5x -30

or 5x +7y = 12

or y =(-5/7)x + 12/7. It's slope is -5/7.

the equation of the line perpendicular has slope 7/5, the negative inverse

y =(7/5)x + b where b is the y intercept

1= (7/5)(1) + b

5 = 7 + 5b

5b = -2

b = -2/5

y = 7x/5 - 2/5 is the line that has all positions exactly half way or equidistant between the 2 birds

None of the choices are half way

A (8,3) B(6,7) C(6,0), D(-5,1)

plug the points into the equation. None work.

3 =?= 7(8)/5 -2/5 = (56-2)/5 = 54/5 = 10.8 which doesn't equal 3

7 =?= 7(6)/5 -2/5 = (42-2)/5 = 8 which doesn't equal 7

0 =?= 7(6)/5-2/5 = 8 which doesn't equal 0

1 =?= 7(-5)/5 -2/5 = -37/5 = -7.4

B (6,7) is almost half way, as (6,8) is on the line.

best answer for coordinates of a birdhouse half way between the birds is (1,1)