Prove that the quadrilateral whose vertices are I(-2,3), J(2,6), K(7,6), and L(3, 3) is a rhombus.
I think in these problems the first step is to express each side as a vector. A vector is the difference between points. When two sides have the same vector (or negatives) it means they're parallel and congruent. So in a rhombus IJKL the vectors IJ and LK should be the same, as should JK and IL. That much assures a parallelogram; we check IJ and JK are congruent to complete the crowing of the rhombus.
Let's calculate these vectors:
IJ = J - I = (2,6) - (-2,3) = (2 - -2, 6 - 3) = (4, 3)
LK = K - L = (7, 6) - (3, 3) = (4, 3)
IJ = LK, so far so good
(Note: If you haven't got to vectors yet you can just show the two sides are the same length, 5, and have the same slope, 3/4, both of which can be read off the vectors.)
JK = K - J = (7,6) - (2,6) = (5,0)
IL = L - I = (3, 3) - (-2, 3) = (5, 0)
Those are the same too.
Now we have to show IJ ≅ JK
The length of IJ is the cliche √4²+3² = 5, the same as JK, so IJ ≅ JK
We showed all four sides are congruent and we have two pair of parallel sides, so we have a rhombus.
