Mid-points of the sides AB and AC of a triangle ABC are ( 3 , 5 ) and ( -3 , -3 ) respectively,then the length of the side BC is

let A = (h,k). Then

B=(h+2(3-h),k+2(5-k))=(6-h,10-k)
C=(h+2(-3-h),k+2(-3,k))=(-6-h,-6-k)

BC^2 = 12^2+16^2 = 340
BC=√340

To find the length of side BC, we need to first find the coordinates of point B and point C.

Since the midpoint of side AB is given as (3, 5), we can use the midpoint formula to find the coordinates of point B. The midpoint formula states that the coordinates of the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) are ((x1 + x2) / 2, (y1 + y2) / 2).

Let's denote the coordinates of point A as (x1, y1) and the coordinates of point B as (x2, y2).

Using the midpoint formula, we have:
(x1 + x2) / 2 = 3 (1)
(y1 + y2) / 2 = 5 (2)

Now, the midpoint of side AC is given as (-3, -3). Denoting the coordinates of point A as (x1', y1') and the coordinates of point C as (x2', y2'), we can use the midpoint formula again:
(x1' + x2') / 2 = -3 (3)
(y1' + y2') / 2 = -3 (4)

Now we have a system of equations from equations (1), (2), (3), and (4) to solve for the coordinates of points B and C.

Solving equations (1) and (3) simultaneously, we get:
x1 + x2 = 6 (5)
x1' + x2' = -6 (6)

Subtracting equation (6) from equation (5), we get:
x1 - x1' + x2 - x2' = 12 (7)

Solving equations (2) and (4) simultaneously, we get:
y1 + y2 = 10 (8)
y1' + y2' = -6 (9)

Adding equation (8) and equation (9), we get:
y1 + y1' + y2 + y2' = 4 (10)

Now we have equations (7) and (10) to solve for the coordinates of points B and C.

Using the fact that the coordinates of point A are (x1, y1), we can rewrite equation (7) as:
x1 + (x2 - x1') - x2' = 12 (11)

Similarly, using the fact that the coordinates of point A are (x1', y1'), we can rewrite equation (10) as:
y1' + (y1 - y2') + y2 = 4 (12)

Equations (11) and (12) can be simplified as follows:
x1 - x1' - x2' = 12 - (x2 - x1') (13)
y1 - y2' + y2 = 4 - (y1 - y2') (14)

Now we have simplified equations (13) and (14) to solve for the coordinates of points B and C.

Since we have two unknowns (x2 - x1') and (y1 - y2'), we need two more equations to solve for them. Fortunately, we have equations (5) and (8) that can be used.

Adding equations (5) and (8), we get:
x1 + x2 + y1 + y2 = 16 (15)

Equation (15) gives us another equation to solve for (x2 - x1') and (y1 - y2'), which we can substitute back into equations (13) and (14).

Once we have the coordinates of points B and C, we can use the distance formula to find the length of side BC.