SOLUTION: Apply Slope, Midpoint and Length Formulas 4. Verify that the quadrilateral with vertices P(2, 3), Q(5, -1), R(10, -1), and S(7, 3) is a rhombus. Can you please help me with this

Question 187157: Apply Slope, Midpoint and Length Formulas
4. Verify that the quadrilateral with vertices P(2, 3), Q(5, -1), R(10, -1), and S(7, 3) is a rhombus.
Can you please help me with this question because i do not know how to verify it. It would be kind if you verify so i can understand it and can do the other question.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
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

. This means that and

Likewise with the point

. This means that and

Start with the distance formula.

Plug in , , , and .

Subtract from to get .

Subtract from to get .

Square to get .

Square to get .

Add to to get .

Take the square root of to get .

So our answer is

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)
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