If A00 B50 Cs4 and Dr4 are the vertices of a rhombus contain

If A(0,0) B(5,0), C(s,4), and D(r,4) are the vertices of a rhombus contained entirely in the first quadrant, determine r and s

Solution

in rhombus length of AB=BC=CD=AD

AB=((5-0)²+(0-0)²) =5

BC=((s-5)²+(4-0)²)=((s-5)²+16)

AD=((r-0)²+(4-0)²)=(r²+16)

BC=AB

=>((s-5)²+16) =5

squaring on both sides

=>((s-5)²+16) =5²

=>(s-5)²+16 =25

=>(s-5)²=9

=>s-5=-3, s-5=3

=>s=5-3 ,s=5+3

=>s=2, s=8

AD=AB

=>(r²+16) =5

=>(r²+16)=5²

=>r²=9

=>r=3

possible value are C(2,4),D(3,4) and  C(8,4),D(3,4)

if  C(2,4),D(3,4) , CD=[(3-2)²+(4-4)²] =1 , but CDAB

if  C(8,4),D(3,4) , CD=[(3-8)²+(4-4)²] =5 , CD=AB

So , r =3,s=8 makes vertices of rhombus