A(h,k) lies on the line y+3x=-10. B lies on the line x+y=4. If the origin is the midpoint of AB, find the values of h and k.

Question

Just a coordinate geometry question that has been evading me for a while. I've tried my best but come up short. Any help will be appreciated

Answer 1

If point A(h,k) lies on the line y = -3x - 10, and its x-coordinate is represented by h, then k = -3h-10. Another way to represent the coordinates of point A: (h, -3h - 10).

Since point B lies on the line y = -x + 4, if we let b represent the x-coordinate of B, then its y-coordinate is -b + 4. A representation for B is (b, -b + 4).

The midpoint of segment AB is the (average of the x-coordinates, average of the y-coordinates).

M = ([ h + b]/2 , [-3h - 10 + (-b + 4)]/ 2 )

M = ( [ h + b]/2 , [-3h - b - 6]/2)

Since both x and y coordinates must be 0 for point M:

h + b = 0 and -3h - b - 6 = 0

b = -h and -3h - b = 6

Substituting:

-3h - (-h) = 6

-2h = 6

h = -3

k = -3(-3) - 10 = -1

So Point A has coordinates (-3, -1).

Since b = -h, b = 3.

That means the coordinates of B must be (3, 1).

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