Let a, b, and c represent positive real numbers. Given the rhombus ABCD with vertices at A(0,0), B(a,0), C(a+b,c) and D(b,c), prove the following theorem.“The diagonals of a rhombus are perpendicular.”
What will be an ideal response?
With vertices as stated, ABCD appears to be parallelogram. For ABCD to further represent a rhombus, 
and 
must be congruent. But 
only if 
(the result of applying the Distance Formula); in turn, 
.
and must be congruent. But only if (the result of applying the Distance Formula); in turn, .Having established the conditions for the rhombus, we will now use the Slope Formula to show that the diagonals are perpendicular. 
so 
. Similarly, 
so 
.
Then the product of the slopes of these diagonals is 
, so 
.
By substitution (recall that ), 
.
With 
, it follows that 
.
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