## 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 $\overline{BD}$.

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 $C$.

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…