Find the area of the quadrilateral OCBAO

In summary: The angles of a circle are 180^0.In summary, you failed to provide enough information to solve the problem. You need to provide more information to solve the problem.
  • #1
chwala
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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.
 
Last edited:
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  • #2
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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  • #3
Nothing in the problem description limits C to a single location.
 
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  • #4
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.
 
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  • #5
...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.
 
  • #6
...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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  • #7
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.
 
Last edited:
  • #8
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".
 
  • #9
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.
 

FAQ: Find the area of the quadrilateral OCBAO

What is the formula to find the area of a quadrilateral?

The area of a quadrilateral can be found using various methods depending on the type of quadrilateral and the available information. A common method is to divide the quadrilateral into two triangles and find the sum of their areas. If the coordinates of the vertices are known, the Shoelace formula can also be used.

How do you use the Shoelace formula to find the area of a quadrilateral?

The Shoelace formula, also known as Gauss's area formula, is used to calculate the area of a polygon when the coordinates of its vertices are known. For a quadrilateral with vertices (x1, y1), (x2, y2), (x3, y3), and (x4, y4), the area is given by: Area = 0.5 * |(x1y2 + x2y3 + x3y4 + x4y1) - (y1x2 + y2x3 + y3x4 + y4x1)|.

What additional information is needed to find the area of quadrilateral OCBAO?

To find the area of quadrilateral OCBAO, you need the coordinates of its vertices O, C, B, A, and O. With these coordinates, you can use the Shoelace formula or divide the quadrilateral into triangles and calculate their areas.

Can the area of a quadrilateral be found using its side lengths alone?

Generally, the area of a quadrilateral cannot be determined solely from its side lengths because the shape and angles between the sides also affect the area. However, if additional information such as the lengths of the diagonals or the internal angles is known, the area can be calculated using formulas like Brahmagupta's formula for cyclic quadrilaterals.

What is Brahmagupta's formula for finding the area of a cyclic quadrilateral?

Brahmagupta's formula is used to find the area of a cyclic quadrilateral (a quadrilateral inscribed in a circle). If the side lengths of the quadrilateral are a, b, c, and d, and the semiperimeter (s) is (a + b + c + d)/2, the area (A) is given by:A = √((s - a)(s - b)(s - c)(s - d)).

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