Thread: 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!

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 different quadrilaterals with diagonals 12 and 8.

Originally Posted by Also sprach Zarathustra

This assumption is wrong. There is different quadrilaterals with diagonals 12 and 8.

Hi!
Thanks for the reply!

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. : )

Originally Posted by Zellator

Hi!
Thanks for the reply!

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.

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.

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!

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!