MHB Show that diagonals in a diamond (rhombus) are orthogonal

AI Thread Summary
The discussion focuses on proving that the diagonals of a rhombus are orthogonal. Participants clarify that the dot product of the diagonals, represented as AC and BD, equals zero, indicating orthogonality. A key point made is that in a rhombus, the lengths of adjacent sides (AB and AD) are equal, which simplifies the calculations. The conversation also highlights the use of vector notation to illustrate the relationships between the sides and diagonals. Ultimately, the proof demonstrates that the diagonals intersect at right angles.
Petrus
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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 $$AC•BD=0$$ cause of dot product and if it's orthogonal the angle is $$\frac{\pi}{2}$$
I understand all the part until the step before the last one
2dqqclx.jpg

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

Regards,
$$|\pi\rangle$$
 
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Re: Show that diagonala in a diamond(romb) is orthogonal

Are you talking about how:

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

If so, this is because:

$$\overline{AD}=\overline{BC}$$
 
Re: Show that diagonala in a diamond(romb) is orthogonal

MarkFL said:
Are you talking about how:

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

If so, this is because:

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

Regards,
$$|\pi\rangle$$
 
Petrus said:
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.
 
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