Question 187157
Remember, a rhombus is an equilateral parallelogram. In other words, a rhombus is a parallelogram in which all of its sides are of the same length.



So all you need to do is find the length of segments PQ, QR, RS, and SP. These lengths should be the same length (in order for the claim to be true). 



I'll show you how to find the length of PQ:


To find the length of PQ, we need to find the distance from point P(2, 3) to Q(5, -1)



So let's use the distance formula



Note: the point *[Tex \LARGE \left(2,3\right)] is *[Tex \LARGE \left(x_{1},y_{1}\right)]. This means that {{{x[1]=2}}} and {{{y[1]=3}}}



Likewise with the point *[Tex \LARGE \left(5,-1\right)] is *[Tex \LARGE \left(x_{2},y_{2}\right)]. This means that {{{x[2]=5}}} and {{{y[2]=-1}}}



{{{d=sqrt((x[1]-x[2])^2+(y[1]-y[2])^2)}}} Start with the distance formula.



{{{d=sqrt((2-5)^2+(3--1)^2)}}} Plug in {{{x[1]=2}}},  {{{x[2]=5}}}, {{{y[1]=3}}}, and {{{y[2]=-1}}}.



{{{d=sqrt((-3)^2+(3--1)^2)}}} Subtract {{{5}}} from {{{2}}} to get {{{-3}}}.



{{{d=sqrt((-3)^2+(4)^2)}}} Subtract {{{-1}}} from {{{3}}} to get {{{4}}}.



{{{d=sqrt(9+(4)^2)}}} Square {{{-3}}} to get {{{9}}}.



{{{d=sqrt(9+16)}}} Square {{{4}}} to get {{{16}}}.



{{{d=sqrt(25)}}} Add {{{9}}} to {{{16}}} to get {{{25}}}.



{{{d=5}}} Take the square root of {{{25}}} to get {{{5}}}.



So our answer is {{{d=5}}} 



So the distance between P(2, 3) and Q(5, -1) is  5 units. 



This consequently means that segment PQ is 5 units long.



Now use the above formula to find the lengths of the other segments (you should get 5 for each length)