Find the coordinate of the point m(0,a) if m is midpoint between A(3,2) and B(b,-4).Find the coordinate of the point m(0,a) if m is midpoint between A(3,2) and B(b,-4).

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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First, we'll write the equation of the line that passes AB.

(xB - xA)/(x-xA) = (yB-yA)/(y-yA)

We'll substitute the coordinates for A and B:

(b-3)/(x-3) = (-4-2)/(y-2)

(b-3)/(x-3) = -6/(y-2)

We'll cross multiply and we'll get:

-6x + 18 = by - 2b - 3y + 6

We'll shift all terms to the left side:

-6x + 18 - by + 2b + 3y - 6 = 0

We'll combine like terms:

-6x + y(-b + 3) + 12 + 2b = 0

If the point (0,a) belongs to the line AB, it's coordinates verify the equation of the line.

-6*0 + a(-b + 3) + 12 + 2b = 0

a(-b + 3) + 12 + 2b = 0

We'll substitute a and b in the expression:

-ab + 3a + 12 + 2b = 0

If (0,a) is the midpoint of AB, then the coordinates of the midpoint are:

xM = (xA+xB)/2

xM = (b+3)/2

But xM = 0

b + 3 = 0

b = -3

yM = (yA+yB)/2

yM = (-4+2)/2

But yB = a

a = -1

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