Let a, b, and c represent positive real numbers. Consider the parallelogram ABCD with vertices at A(0,0), B(a,0), C(a+b, c), and D(b,c). In order that ABCD further represents a rhombus, prove that 
.[Note: No drawing provided.]
What will be an ideal response?
For ABCD to represent a rhombus, the lengths of 2 adjacent sides of 
must be congruent. Of course, 
when 
. With A and D on the x axis, we see that 
. Applying the Distance Formula, we also have 
or 
.
must be congruent. Of course, when . With A and D on the x axis, we see that . Applying the Distance Formula, we also have or .Thus, ABCD is a rhombus when 
.
Mathematics
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Mathematics