P is the mid point of A(-1,-1);B(-1,4)P(x,y)=x₁+x₂/2 , y₁+y₂/2here x₁=-1, x₂=-1y₁=-1, y₂=4P(x,y) = -1+(-1)/2 and -1+4/2=-1-1/2 and 3/2=-1 and 1.5∴P(-1, 1.5)

Q is the mid point of B(-1,4);C(5,4)Q(x,y)=x₁+x₂/2 , y₁+y₂/2= -1+5/2 , 4+4/2=4/2 , 8/2=2,4∴Q(2,4)

R is the mid point of C(5,4)D(5,-1)R(x,y) = 5+5/2 , 4+(-1)/2=10/2,4-1/2=5,3/2=5,1.5∴R(5,1.5)

S is the mid point of A(-1,-1) ;D(5,-1)S(x,y)= -1+(-1)/2+5+(-1)/2=-1-1/2, 5-1/2=-2/2 , 4/2=-1,2∴S(-1,2)

PQRS is a quadrilateral but to prove what type of quad. it is use the distance formulaP(-1, 1.5) ; Q(2,4)PQ=√[(x₂-x₁)²+(y₂-y₁)²]PQ =√(2-(-1))²+(4-1.5)²=√(3)²+(2.5)²=√9+6.25=√15.25∴ PQ =√15.25

Q(2,4); R(5,1.5)QR =√[(5-2)²+(1.5-4)²]=√(3)²+(-2.5)²=√9+6.25=√15.25∴QR =√15.25

R(5,1.5) ; S(-1,2)RS=√[(5+1)²+(2-1.5)²]= √6²+(0.5)²=√36+0.25=√36.25∴RS=√36.25

S(-1, 2) ;P(-1,1.5)SP=√[(-1+1)²+(1.5-2)²]=√(0)²+(0.5)²=√0+0.25=√0.25=0.5∴SP = 0.5

On comparing all sides joined by the midpoints of the rectangle we find that PQRS is just a quadrilateral not any type of quad.