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Circle ratio problem!

I need some help in answering this Q

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(edited 4 years ago)
Reply 1
Original post by Yatayyat
I need some help in answering this Q

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What have you tried? Please post your working.
Reply 2
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]

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?
(edited 4 years ago)
Reply 3
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]

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=2:7S1 : S2 = 2:7

becomes

S1S2=27\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.
Reply 4
Original post by Notnek
You can change ratio equations into algebraic equations:

S1:S2=2:7S1 : S2 = 2:7

becomes

S1S2=27\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?
My name is michalwithab
Reply 6
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.
Reply 7
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 :smile:

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