Proof that The Diagonals of a Rhombus are Perpendicular

In the previous post, we have proved the converse of the Pythagorean Theorem.  In this post, we will prove that the diagonals of a rhombus are perpendicular to each other. That is, if we have parallelogram ABCD with diagonal \overline{AC} and \overline{BD}, then \overline{AC} is perpendicular to \overline{BD}.

parallelogram 2

What We Know

A rhombus is a parallelogram with four congruent sides. So, all sides of rhombus ABCD are congruent. That is

\overline{AB} \cong \overline{BC} \cong \overline{CD} \cong \overline{DA}Continue reading…

 

Proof of the Converse of the Pythagorean Theorem

The Pythagorean Theorem states that if ABC is a triangle right angled at C, then,

a^2 + b^2 = c^2.

The converse of the Pythagorean Theorem states that if

a^2 + b^2 = c^2 holds,

then triangle ABC is a right triangle right angled at C.

right triangle

This means that in order to prove the converse of the Pythagorean Theorem, we need to prove that \angle C in the figure above is a right angle.  Now, we discuss the proof. Continue reading…

 

Absolute Value Proofs: Products and Quotients

The absolute value of a number is its distance from 0. Thus, all absolute values are either positive or 0. That is, if we have a number x, then the absolute value of x which can be written as |x| is equal to

(a) x if x is positive

(b) -x if x is negative

(c) 0 if x is 0.

Also, for any real number, x^2 is positive or 0 (if x = 0).  Therefore, \sqrt{x^2} is

(a) x if x is positive

(b) -x if x is negative

(c) 0 if x is 0.

If the -x part is a bit confusing, consider take for example  Continue reading…

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