Find the area of the quadrilateral OCBAO

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The discussion revolves around determining the angles of quadrilateral OCBAO, with the original poster attempting to derive equations based on angle relationships. They initially miscalculated angles and recognized the need for corrections, including the potential use of the cosine rule. Other participants highlighted the ambiguity in the position of point C on the circle, emphasizing that without specific constraints, the problem cannot be definitively solved. The importance of understanding circle properties and the relationships between angles subtended by chords was also stressed. Ultimately, the original poster acknowledged mistakes and expressed the need for further analysis to clarify the situation.
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Homework Statement
This is my own question. I just made some extension to original question( see diagram)
Relevant Equations
understanding of circle properties.
1689427506107.png


My challenge was on trying to determine the angles: My approach;

1689427578120.png


came up with a number of equations: ie

##m+n=70^0##
##r=p+40^0##
##q-2r=100^0, ⇒ r=50^0 + \dfrac{1}{2} q##

then it follows that,
##2q+100^0=180^0##
##⇒q=40^0, r=70^0, p=m=30^0, n=40^0##

##m+40^0+t=180^0, ⇒t=110^0##

and
##q+p+s=180^0##
##40+30+s=180^0, s=110^0##

problem here...i will need to check on this...

I need to have ##t+s=180^0##.

I know once i am certain on the angles then finding area is as easy as abc...

i see my own mistake...i will go through this again...I may need to use cosine rule...coming back in a moment.
 
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chwala said:
Homework Statement: This is my own question. I just made some extension to original question( see diagram)
Relevant Equations: understanding of circle properties.

View attachment 329308

My challenge was on trying to determine the angles: My approach;
. . .

i see my own mistake...i will go through this again...I may need to use cosine rule...coming back in a moment.

It looks like you failed to provide enough information.
 
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Nothing in the problem description limits C to a single location.
 
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As already noted by @SammyS and @Frabjous, the position of point C on the circle needs to be defined (unless maybe AC and OB are meant to be perpendicular?).

Your hand-drawn Post #1 diagram is wrong. ∠ACB is not 40º.

∠ACB is the angle subtended by AB at a point C on the circumference.
∠AOB is the angle subtended by AB at the centre.
There is a simple relation between these two angles but they are not equal.
 
...I realised that the question has many unknowns...been looking at it for last 30 minutes...it cannot be solved...and true ##C## can lie at any point on the circle circumference.
 
...supposing we are told that the length AD = DC... so that we have the point C fixed at a point. Are we going to have some breakthrough? i need to analyse this later...

1689443439374.png
 
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While you added a condition, you also added a new unknown. While for a given D, C is determined, D is not limited to a single point.

You are too eager to jump into the analysis, instead of sitting back and developing a strategy.
 
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chwala said:
...supposing we are told that the length AD = DC... so that we have the point C fixed at a point. Are we going to have some breakthrough? i need to analyse this later...

View attachment 329311

The equations that i had last were:
##2r+q=180^0##
##4x-q=40^0##
As pointed out by @Steve4Physics "∠ACB is not 40º".
 
SammyS said:
As pointed out by @Steve4Physics "∠ACB is not 40º".
It is 40^0 check on the circle properties... angles subtended by the same chord....

aaaargh its ##20^0## ...you are correct.
 

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