Condition on sides of a quadrilateral and some diagonals

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Condition on sides of a quadrilateral and some diagonals

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inequalities proposed

rearrangement inequality

geometry proposed

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Source: RMO (Mumbai Region) 2015 P1

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#1 h Dec 6, 2015, 12:52 PM • 2 Y

Y by Adventure10, Mango247

Let $ABCD$ be a convex quadrilateral with $AB=a$ , $BC=b$ , $CD=c$ and $DA=d$ . Suppose
$a^2+b^2+c^2+d^2=ab+bc+cd+da,$ and the area of $ABCD$ is $60$ sq. units. If the length of one of the diagonals is $30$ units, determine the length of the other diagonal.

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#2 h Dec 6, 2015, 1:42 PM • 1 Y

Y by Adventure10

are you sure you typed it correctly.

as $a^2+b^2+c^2+d^2=ab+bc+cd+da \Rightarrow (a-b)^2+(b-c)^2+(c-d)^2+(d-a)^2=0$ or $a=b=c=d$ .

edit = hence quad. must be rhombus now $area = d1.d2/2$ which gives $d2=4$ .

This post has been edited 2 times. Last edited by aditya21, Dec 6, 2015, 2:22 PM
Reason: e

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#3 h Dec 6, 2015, 1:45 PM • 1 Y

Y by Adventure10

aditya21 wrote:

are you sure you typed it correctly.

as $a^2+b^2+c^2+d^2=ab+bc+cd+da \Rightarrow (a-b)^2+(b-c)^2+(c-d)^2+(d-a)^2=0$ or $a=b=c=d$ .

thus $a^2=60$ but than we doesnt get length of diagonal as $30$ .

Try rechecking your logic.

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#4 h Dec 6, 2015, 2:05 PM • 2 Y

Y by GoJensenOrGoHome, Adventure10

aditya21 wrote:

are you sure you typed it correctly.

as $a^2+b^2+c^2+d^2=ab+bc+cd+da \Rightarrow (a-b)^2+(b-c)^2+(c-d)^2+(d-a)^2=0$ or $a=b=c=d$ .

thus $a^2=60$ but than we doesnt get length of diagonal as $30$ .

I think this means that it's a rhombus; by the diagonal formula for area, the other diagonal should be $\boxed{4}$ ...

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#5 h Dec 6, 2015, 2:21 PM • 3 Y

Y by sarthak786, Adventure10, Mango247

yeah i was stupid to think it as square.

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#7 h Aug 15, 2019, 10:56 AM • 3 Y

Y by AlastorMoody, Adventure10, Mango247

And algebra problem disguised as Geometry
$a^2+b^2+c^2+d^2=ab+bc+cd+da\Rightarrow a=b=c=d$ by AM-GM inequality.
Thus, it is a rhombus.
Area of rhombus is $\frac{d_1d_2}{2}$
So, the other diagonal has length 4

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#8 h Jul 9, 2020, 3:17 PM

Y by

YESMAths wrote:

Let $ABCD$ be a convex quadrilateral with $AB=a$ , $BC=b$ , $CD=c$ and $DA=d$ . Suppose
$a^2+b^2+c^2+d^2=ab+bc+cd+da,$ and the area of $ABCD$ is $60$ sq. units. If the length of one of the diagonals is $30$ units, determine the length of the other diagonal.

Simple.. we may also use rearrangement inequality for this one.. a one-liner !

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