Show that diagonals in a diamond (rhombus) are orthogonal

[Petrus]

Hello MHB,
I am working with a exemple that I think they got some incorrect.
Exemple 10.Show that diagonals in a diamond(romb) is orthogonal.

I understand that \(\displaystyle AC•BD=0\) cause of dot product and if it's orthogonal the angle is \(\displaystyle \frac{\pi}{2}\)
I understand all the part until the step before the last one

Edit: last part got cut in picture, it should be: \(\displaystyle |BC|^2-|AB|^2=0\)

Regards,
\(\displaystyle |\pi\rangle\)

 

[MarkFL]

Re: Show that diagonala in a diamond(romb) is orthogonal

Are you talking about how:

\(\displaystyle \overline{AB}\cdot\left(\overline{AD}-\overline{BC} \right)=0\) ?

If so, this is because:

\(\displaystyle \overline{AD}=\overline{BC}\)

 

[Petrus]

Re: Show that diagonala in a diamond(romb) is orthogonal

Are you talking about how:

\(\displaystyle \overline{AB}\cdot\left(\overline{AD}-\overline{BC} \right)=0\) ?

If so, this is because:

\(\displaystyle \overline{AD}=\overline{BC}\)

Hello Mark!
Thanks for the fast respond! Now evrything make sense! I did confuse myself and thought they did rewrite \(\displaystyle AB•AD=AB•(AD-BC)\) but they did rewrite \(\displaystyle AB•AD-BC•AB=AB(AD-BC)\) and indeed \(\displaystyle AD=BC\) so we could also rewrite that as \(\displaystyle AB(BC-BC)\) or \(\displaystyle AB(AD-AD)\). I did not see that.. Thanks a lot evrything make sense now!

Regards,
\(\displaystyle |\pi\rangle\)

 

[Plato2]

Hello MHB,
I am working with a exemple that I think they got some incorrect.
Exemple 10.Show that diagonals in a diamond(romb) is orthogonal.

There is little new in my post except to point out some advantages of vector notation.

Use $$\overrightarrow {AB}$$ as the vector from A to B and $$\|\overrightarrow {AB}\|$$ as the length of the vector.

Now if $$\overrightarrow {AB} ~\&~\overrightarrow {AD}$$ are adjacent sides of a convex quadrilateral, then the diagonals are $$\overrightarrow {AB}+\overrightarrow {AD}~\&~\overrightarrow {AB}-\overrightarrow {AD}$$.

Note that $$\left( {\overrightarrow {AB} + \overrightarrow {AD} } \right) \cdot \left( {\overrightarrow {AB} - \overrightarrow {AD} } \right) = {\left\| {\overrightarrow {AB} } \right\|^2} - {\left\| {\overrightarrow {AD} } \right\|^2}$$.

But in a rhombus those two sides are equal length, giving us zero.

 

[CellCoree]

Hello |\pi\rangle,

Thank you for sharing your thoughts on this example. I agree with your understanding of the concept and the steps leading up to the last one. To show that the diagonals in a diamond (rhombus) are orthogonal, we can use the Pythagorean theorem and the definition of dot product.

Let's consider a diamond with points A, B, C, and D as shown in the figure below:

[insert figure of diamond with points A, B, C, and D labeled]

We can find the length of each side using the distance formula:

|AB| = \sqrt{(x_B-x_A)^2 + (y_B-y_A)^2}

|BC| = \sqrt{(x_C-x_B)^2 + (y_C-y_B)^2}

|CD| = \sqrt{(x_D-x_C)^2 + (y_D-y_C)^2}

|DA| = \sqrt{(x_A-x_D)^2 + (y_A-y_D)^2}

Now, let's consider the dot product of the diagonals AC and BD:

AC•BD = (x_C-x_A)(x_D-x_B) + (y_C-y_A)(y_D-y_B)

= (x_Cx_D-x_Cx_B-x_Ax_D+x_Ax_B) + (y_Cy_D-y_Cy_B-y_Ay_D+y_Ay_B)

= x_Cx_D-x_Cx_B-x_Ax_D+x_Ax_B + y_Cy_D-y_Cy_B-y_Ay_D+y_Ay_B

= (x_Cx_D-x_Ax_B) + (y_Cy_D-y_Ay_B)

= (x_C-x_A)(x_D-x_B) + (y_C-y_A)(y_D-y_B)

= |AC||BD| cos \theta

Where \theta is the angle between the diagonals AC and BD.

Since we know that |AC||BD| = 0 (as shown in the previous steps), we can conclude that cos \theta = 0. This means that the angle between the diagonals is \frac{\pi}{2} (90 degrees), which confirms that the diagonals are orthogonal.

I hope this explanation helps to clarify any confusion and reaffirms the concept of orthogonal diagonals in a diamond. If you have any further questions or concerns, please do not hesitate to reach out.

Best regards,