-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."

Question

Quizplus · Accepted Answer

The Answer is With vertices as stated, ABCD appears to be parallelogram. For ABCD to further represent a rhombus,

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, . 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 .

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

.

-Let A, B, and C Represent Positive Real Numbers