Let Q = (x, -x + 6)
So....by the Pythagorean Theorem...
[Distance from (0,0) to Q]^2 + [ Distance from (10, -10) to Q ]^2 = [Distance from (0,0) to (10, -10)]^2
[ (x )^2 + ( -x + 6)^2] + [ (10-x)^2 + (-10 - (-x + 6)*^2 ] = 200
[ (x^2) + (-x + 6)^2] + [ (10 - x)^2 + ( x - 16)^2] = 200
x^2 + x^2 - 12x + 36 + x^2 - 20x + 100 + x^2 - 32x + 256 = 200
4x^2 - 64x + 192 = 0 divide through by 4
x^2 - 16x + 48 = 0 factor
(x - 12) ( x - 4) = 0
Set each factor to 0, solve for x and we get that x = 12 or x = 4
When x = 4, y = 2
When x = 12, y = -6
So...Q = (4, 2) or Q = ( 12, - 6)
So....the sum of the coordinates = 12
Here's a graph :