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jacks

Guest

max. area of quadrilateral

what is the maximum area of a cyclic quadrilateral whose one angle is 120 degree

Bob

Administrator

Registered: 2010-06-20

Posts: 10,583

Re: max. area of quadrilateral

hi jacks,

From a diagram in Sketchpad, it's looking like it will be when the other two angles are both 90.

Haven't proved it yet though. 

I'll post again if I manage it.

edit:

Mark A, B  and C on a circle so that ABC = 120

Let D be a point on the circle.

The area of ACD is determined by the base AC and the height of the triangle, determined by where D is on the circle.  To maximise this, move D until it is diametrically opposite to B.

If BD is a diameter, then  A = C = 90.

RE-EDIT Final version.

(i)  Draw the circle, centre E.

(ii)  Choose (with no loss of generality) a point A on the circumference.

(iii) C is now fixed because we want ABC = 120, so the angle at the centre AEC = 240.  So rotate the radius EA around E until AEC = 240, and make this point on the circumference C.  (Note:  any point B on the major arc will have the desired angle property since, for all points B, ABC = half AEC.)

(iv)  Triangle ABC has area depending on its base, AC,  and its perpendicular height.  So construct a line, perpendicular to AC to go through E.  Continue this line until it cuts the circle.  Make this point B.  This will give the maximum area for ABC.

(v)  Produce BE across the circle to cut again at D.  This will give the maximum area for ADC.  (Note that BD is still a diameter as stated in my first edit so the angles will still be 90.  But this way of choosing B maximises the whole shape.

The resulting shape is a kite.

Bob

Last edited by Bob (2011-11-14 05:29:06)

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Children are not defined by school ...........The Fonz
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Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: max. area of quadrilateral

Hi all;

The maximum value is achieved as Bob's work shows when the other three angles are 60° , 90° and 90° . This can by shown by calculus for any cyclic quadrilateral with one angle of 120°  the maximum value is:

Where r is the radius of the circumscribed circle.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.