Two sides of the rectangle must be perpendicular—but which two? The product of the slopes of these lines must be -1 to be perpendicular. Label the points: A(-5,3), B(1,-1), C(-1,-4). D is the vertex whose coords we have to find.

Slope of AB=(3-(-1))/(-5-1)=-4/6=-⅔.

Slope of BC=(-1-(-4))/(1-(-1))=3/2.

Slope of AC=(3-(-4))/(-5-(-1))=-7/4.

Clearly AB and BC are perpendicular because (-2/3)(3/2)=-1.

So DC is parallel to AB and perpendicular to AD.

B is as far over to the right from A as D is to the left of C (x direction).

B is as far below A as D is above C (y direction).

B is 1-(-5)=6 units right of A and 3-(-1)=4 units below A.

D is 6 units left of C at x=-1-6=-7.

D is 4 units above D at y=-4+4=0.

D is (-7,0), answer A.