# Proving type of quadrilateral by its diagonal lengths

• Jun 11th 2011, 12:58 PM
Zellator
Proving type of quadrilateral by its diagonal lengths
Hi Forum

Is there an easy and objective way to find the type of quadrilateral just by its diagonal lengths? I mean, with no calculation.
If we were to find the area of a quadrilateral with diagonals 12 and 8. Having the diagonals it's not enough to find the area.
Only if the quadrilateral is a rhombus, but we can't just assume that it is a rhombus. Since there is no more information given, like bisection of diagonals and angles.

Thanks!
• Jun 11th 2011, 01:27 PM
Also sprach Zarathustra
Quote:

Originally Posted by Zellator
Hi Forum

Is there an easy and objective way to find the type of quadrilateral just by its diagonal lengths? I mean, with no calculation.
If we were to find the area of a quadrilateral with diagonals 12 and 8. Having the diagonals it's not enough to find the area.
Only if the quadrilateral is a rhombus, but we can't just assume that it is a rhombus. Since there is no more information given, like bisection of diagonals and angles.

Thanks!

This assumption is wrong. There is $\aleph$ different quadrilaterals with diagonals 12 and 8.
• Jun 11th 2011, 01:54 PM
Zellator
Quote:

Originally Posted by Also sprach Zarathustra
This assumption is wrong. There is $\aleph$ different quadrilaterals with diagonals 12 and 8.

Hi!

This is the question
The diagonals of the quadrilateral LEAK have diagonals LA=12 and EK=8. Find the maximum area this quadrilateral can have under these conditions
In the answer is just assumed that LEAK is a rhombus and the answer is 1/2xLAxEK

If we are after the maximum area is it safe to simple use a rhombus?
Is it because of its shape that the area will be maximized?
Sorry I left that bit of information out. I see it was actually important. : )
• Jun 11th 2011, 02:30 PM
Also sprach Zarathustra
Quote:

Originally Posted by Zellator
Hi!

This is the question
The diagonals of the quadrilateral LEAK have diagonals LA=12 and EK=8. Find the maximum area this quadrilateral can have under these conditions
In the answer is just assumed that LEAK is a rhombus and the answer is 1/2xLAxEK

If we are after the maximum area is it safe to simple use a rhombus?
Is it because of its shape that the area will be maximized?
Sorry I left that bit of information out. I see it was actually important. : )

You can prove using calculus tools that maximum area of quadrilateral is when the diagonals are perpendicular.
• Jun 11th 2011, 03:20 PM
Also sprach Zarathustra
We can prove that maximum area of quadrilateral is when the diagonals are perpendicular, by proving that for triangles. If ABC is triangle and x is the angle between a and b then S, the area of ABC is give by: (1/2)*a*b*sin(x).
max{S}=max{ (1/2)*a*b*sin(x) }=(1/2)*a*b*max{sin(x)}=(1/2)*a*b*1 (Why?), hence sin(x)=1 ==> x=/pi/2.
• Jun 11th 2011, 06:36 PM
Zellator
Quote:

Originally Posted by Also sprach Zarathustra
We can prove that maximum area of quadrilateral is when the diagonals are perpendicular, by proving that for triangles. If ABC is triangle and x is the angle between a and b then S, the area of ABC is give by: (1/2)*a*b*sin(x).
max{S}=max{ (1/2)*a*b*sin(x) }=(1/2)*a*b*max{sin(x)}=(1/2)*a*b*1 (Why?), hence sin(x)=1 ==> x=/pi/2.

Amazing! Yes I agree!
Clever resolution for it. It is a calculus question really. So we should use calculus.
Thanks for your time Also sprach Zarathustra!
That really clears a lot of things up.
All the best!
• Jun 11th 2011, 06:40 PM
Also sprach Zarathustra
Quote:

Originally Posted by Zellator
Amazing! Yes I agree!
Clever resolution for it. It is a calculus question really. So we should use calculus.
Thanks for your time Also sprach Zarathustra!
That really clears a lot of things up.
All the best!

Thank you!