Circle ratio problem!

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Yatayyat
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#1
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#1
I need some help in answering this Q

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Notnek
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#2
(Original post by Yatayyat)
I need some help in answering this Q

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What have you tried? Please post your working.
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Yatayyat
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(Original post by Notnek)
What have you tried? Please post your working.
Given that I can use

Area of a triangle: 1/2 ab sin theta
And that area of a sector: 1/2 x r^2 x theta

Hence I know that:

Minor sector area AOB: 1/2 x r^2 x theta
Plus triangle AOB = 1/2 x r^2 x theta [given that side length is r so a = b in this case]

This means that area S1 is: 1/2 x r^2 x theta - 1/2 x r^2 x sin theta

But how would I use the fact that S1 : S2 = 2 : 7 to help give me the required expression that I need to somehow show?
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Notnek
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(Original post by Yatayyat)
Given that I can use

Area of a triangle: 1/2 ab sin theta
And that area of a sector: 1/2 x r^2 x theta

Hence I know that:

Minor sector area AOB: 1/2 x r^2 x theta
Plus triangle AOB = 1/2 x r^2 x theta [given that side length is r so a = b in this case]

This means that area S1 is: 1/2 x r^2 x theta - 1/2 x r^2 x sin theta

But how would I use the fact that S1 : S2 = 2 : 7 to help give me the required expression that I need to somehow show?
You can change ratio equations into algebraic equations:

S1 : S2 = 2:7

becomes

\displaystyle \frac{S1}{S2} = \frac{2}{7}

S2 is just the full circle minus S1 so you should be able to carry on from here.
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Yatayyat
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#5
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(Original post by Notnek)
You can change ratio equations into algebraic equations:

S1 : S2 = 2:7

becomes

\displaystyle \frac{S1}{S2} = \frac{2}{7}

S2 is just the full circle minus S1 so you should be able to carry on from here.
So then area of complete circle is pi x r^2

So S2 = S1 - pi x r^2

Could I say that whole circle is same as saying 9 equal parts
Since S1 is 2 parts and S2 is 7 parts, therefore S1 + S2 = whole circle = 9 parts

A new ratio I thought can be S1 : complete circle = 2 : 9?

Implying that S1 = 2/9 * complete circle area

So earlier on S1 = 1/2 x r^2 x theta - 1/2 x r^2 x sin theta
But S1 is also 2/9 x pi x r^2

I can equate the two giving:

1/2 x r^2 x theta - 1/2 x r^2 x sin theta = 2/9 x pi x r^2

Cancelling out the r^2's that are common on both sides:

1/2 x theta - 1/2 sin theta = 2/9 pi

Then multiplied both sides by 2:

theta - sin theta = 4/9 pi

Subtracting 4/9 pi on both sides:

theta - sin theta - 4/9 pi = 0

So finally I got it in the form they wanted. Just wondering could I have done it another way by comparing the area of S2 with area of complete circle since I didn't actually use the area of S2 here?
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Blueclueless
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My name is michalwithab
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Notnek
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#7
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#7
(Original post by Yatayyat)
So then area of complete circle is pi x r^2

So S2 = S1 - pi x r^2

Could I say that whole circle is same as saying 9 equal parts
Since S1 is 2 parts and S2 is 7 parts, therefore S1 + S2 = whole circle = 9 parts

A new ratio I thought can be S1 : complete circle = 2 : 9?

Implying that S1 = 2/9 * complete circle area

So earlier on S1 = 1/2 x r^2 x theta - 1/2 x r^2 x sin theta
But S1 is also 2/9 x pi x r^2

I can equate the two giving:

1/2 x r^2 x theta - 1/2 x r^2 x sin theta = 2/9 x pi x r^2

Cancelling out the r^2's that are common on both sides:

1/2 x theta - 1/2 sin theta = 2/9 pi

Then multiplied both sides by 2:

theta - sin theta = 4/9 pi

Subtracting 4/9 pi on both sides:

theta - sin theta - 4/9 pi = 0

So finally I got it in the form they wanted. Just wondering could I have done it another way by comparing the area of S2 with area of complete circle since I didn't actually use the area of S2 here?
If the area of S1 : S2 = 2:7 then that means that S1:Whole circle = 2 : 9

So you could do it without S2 in your working if you like but it's pretty similar.
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Yatayyat
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#8
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#8
(Original post by Notnek)
If the area of S1 : S2 = 2:7 then that means that S1:Whole circle = 2 : 9

So you could do it without S2 in your working if you like but it's pretty similar.
Oh okay thanks! Just happy that it's along the right lines
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