As you probably remember the area S of a triangle of which the three vertices coordinates are given equal to one half of the determinant formed with them as follows:

X1 Y1 1
X2 Y2 1 = 2*S
X3 Y3 1

Then you already got the coordinate of one the vertex (-7, 5) . To obtain the other two, remember since the straight line must pass through the origin both vertices must comply with the equation y = mx besides each given equation, where m is the unknown

So by substitution of y = mx in the two original equations you get :

X2 = 12/(m-1) and Y2 = 12m/(m-1) for eq. y = x +12

And

X3 = -9/(m+2) and Y2 = -9m/(m+2) for eq. y = -2x -9

Plugging these coordinates in the determinant above and solving it

After an easy simplification you get a second degree equation in m as follows

50 m2 + 71m + 23 = 0 (m2 means the square of m)

Solving it you get the two solution

m = -1/2 and m= -23/25

I let you to have check the answers by graphic and algebra, I already have done it