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#1 2013-12-30 01:50:03

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
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Circles

1. If an isosceles triangle PQR, in which PQ=PR=12 cm, is inscribed in a circle of radius 18 cm, find the area of the triangle


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
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#2 2013-12-30 02:26:22

Nehushtan
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Registered: 2013-03-09
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Re: Circles


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#3 2013-12-30 02:41:29

Agnishom
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From: Riemann Sphere
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Re: Circles

Could you please post the workings?

P.S.: That is correct


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#4 2013-12-30 05:27:58

atran
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Registered: 2013-07-12
Posts: 91

Re: Circles

The problem becomes easy when you draw three radii inside the triangle; each radius connected to one point of the triangle.

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#5 2013-12-30 05:28:49

Agnishom
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From: Riemann Sphere
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Re: Circles

Thanks for helping. I've drawn the radii. What next?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#6 2013-12-30 05:29:14

atran
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Registered: 2013-07-12
Posts: 91

Re: Circles

Use the cosine rule first, and then the sine rule.

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#7 2013-12-30 05:43:28

Agnishom
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From: Riemann Sphere
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Re: Circles

I am allowed to use the pythagoras theorem which is a stripped version of the cosine rule


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#8 2013-12-30 05:56:38

atran
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Registered: 2013-07-12
Posts: 91

Re: Circles

Aren't you allowed to use the cosine rule?

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#9 2013-12-30 05:58:06

Agnishom
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From: Riemann Sphere
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Re: Circles

possibly not; but I would want to see a solution with it anyway


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#10 2013-12-30 06:32:20

Nehushtan
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Registered: 2013-03-09
Posts: 957

Re: Circles

atran wrote:

The problem becomes easy when you draw three radii inside the triangle; each radius connected to one point of the triangle.

You can’t. The sides of the triangle are shorter than the radius of the circle.

Agnishom wrote:

I would want to see a solution with it anyway


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#11 2013-12-30 06:39:03

atran
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Registered: 2013-07-12
Posts: 91

Re: Circles

Nehushtan wrote:
atran wrote:

The problem becomes easy when you draw three radii inside the triangle; each radius connected to one point of the triangle.

You can’t. The sides of the triangle are shorter than the radius of the circle.

You're right.

Last edited by atran (2013-12-30 06:39:16)

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#12 2013-12-30 06:50:16

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Circles

Hi;


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.

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#13 2013-12-30 07:10:36

atran
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Registered: 2013-07-12
Posts: 91

Re: Circles

Here is my solution. I hope I haven't made mistakes. If I have, please correct me.
Here is the link to the image:

http://imgur.com/JdjngAY

Look at the triangle PCR; you know all the sides, so you can use the cosine rule to find the angle (a). After a little work you have:

Now you need (x). To find it, you need to use the sine rule. Applying the rule you get:

Next you need to find the height of the original triangle (call it h):

So the area is equal to:

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#14 2013-12-30 07:18:49

anonimnystefy
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From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: Circles

And sin(arccos(x))=sqrt(1-x^2).


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#15 2013-12-30 14:41:13

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: Circles

Is there no way that the sum can be done using no coordinate geometry or trigonometry?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#16 2013-12-30 14:59:37

Nehushtan
Member
Registered: 2013-03-09
Posts: 957

Re: Circles

Agnishom wrote:

Is there no way that the sum can be done using no coordinate geometry or trigonometry?

You certainly can do it that way

In Atran’s diagram, produce PC to the other end of the diameter at S. Also let D be the midpoint of QR.

AgnishomsCircleProblem-1.png

Then SQP is a right angle so

Also the triangles SQP and QDP are similar. Thus

from which you can proceed to calculate PD and hence the area of triangle PQR.

Last edited by Nehushtan (2013-12-30 15:02:16)


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#17 2013-12-30 15:17:43

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: Circles

Thanks, this is what I was looking for


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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