# The point M(a, 1) is the midpoint of the line segment VW with point V(-10, b) and point W(6, 11) . Determine the value of a + b .

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M(a, 1) is a mid point for the segment VW.

V(-10,b) and W(6,11)

We know that the mid point between two point (x1,y1) and (x2,y2) is:

((x1+x2)/2 , (y1+y2)/2)

Then :

a= (-10 + 6)/2 ==> 2a= -4 ==> a= -2

1= (11+b)/2 ==> 11+b =2 ==> b= -9

Then the point M(-2,1) is a midpoint of vw where v(-10,-9) and w(6,11)

To check:

(-2,1) = ( -10+6)/2 , (-9+11)/2)

(-2,1) = (-4/2 , 2/2)

(-2,1) = (-2,1)

The coordinates of the midpoint M of the line VW, could be found using the formula:

xM = (xV+xW)/2

xM = (-10+6)/2

xM = -2

yM = (yV+yW)/2

yM = (b+11)/2

But, from enunciation, we know that the coordinates of M are:

xM = a and yM = 1

If xM = -2, then** a = -2**

If yM = (b+11)/2, then 1 = (b+11)/2

b+11 = 2

We'll subtract 11 both sides, to isolate b to the left side:

b = 2-11

**b = -9**

Now, knowing a and b, we could find the sum:

a+b = -2-9

**a+b = = -11**

We use the midpoint formula for x and y coordinates and substitite the given given coordinates and solve for the indetrminates a and b from the two equations and then find the result for a+b.

The coordinates M(x, y) of the mid points are given by:

Mx = (Vx+Wx)/2 and My = (Vy+Wy)/2.

Given M(x,y) = M(a,1) and V(x,y) = V(-10,b) and W(x,y) = W(6,11).

a = (-10+6)/2 = -4/2 = -2.

1 = (b+11)/2. Or 2 = b+11. Or b = 2-11 = -9.

So a+b = -2-9 = -11

Coordinates of the mid point of a line joining two points (x1, y1) and x2, y2) is given by:

Mid point --> [(x1 + x2)/2, (y1 + y2)/2]

Using the above equation we calculate coordinates of the mid point M of VW as follows and equate with their given values as follows:

x-coordinate:

a = (-10 + 6)/2 = -2

y-coordinate:

1 = (b + 11)/2

Multiplying both sides of equation by 2:

2 = b + 11

b = 11 - 2 = -9

Next we calculate the value of (a + b) substituting the values of a and b calculated above:

a + b = -2 - 9 = -11