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

Algebra ->  Quadratic Equations and Parabolas -> 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       Log On


   

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) About Me  (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 is . This means that x%5B1%5D=2 and y%5B1%5D=3


Likewise with the point is . This means that x%5B2%5D=5 and y%5B2%5D=-1


d=sqrt%28%28x%5B1%5D-x%5B2%5D%29%5E2%2B%28y%5B1%5D-y%5B2%5D%29%5E2%29 Start with the distance formula.


d=sqrt%28%282-5%29%5E2%2B%283--1%29%5E2%29 Plug in x%5B1%5D=2, x%5B2%5D=5, y%5B1%5D=3, and y%5B2%5D=-1.


d=sqrt%28%28-3%29%5E2%2B%283--1%29%5E2%29 Subtract 5 from 2 to get -3.


d=sqrt%28%28-3%29%5E2%2B%284%29%5E2%29 Subtract -1 from 3 to get 4.


d=sqrt%289%2B%284%29%5E2%29 Square -3 to get 9.


d=sqrt%289%2B16%29 Square 4 to get 16.


d=sqrt%2825%29 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)