Math Is Fun Forum

math9maniac

Member

From: Tema

Registered: 2015-03-30

Posts: 443

Circle problem

Hi everyone. I'm so busy with school these days it seems it's been ages since I was here. Hope y'all doing great. Could you please help out with this question? Thanks.

A student drew two circles in a diagram such that one is inscribed in the other. The smaller circle touches the larger circle and also touches the diameter AB of the larger circle at P.
If |AP| = 22 cm and |PB| =10 cm, calculate the radius of the smaller circle.

I've never before come across such a question and I have no idea how to go about it. The answer with explanation will be appreciated. Thanks.

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Only a friend tells you your face is dirty.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Circle problem

hi math9maniac

When a smaller circle touches a larger one internally (at E in my diagram) they have a common tangent and hence their centres lie on the same line (EDC in my diagram, where C is the centre of the larger and D of the smaller).

You know AP and PB so you know the diameter AB and hence the radius CE = CA.  You can also work out CP.

So if you call the smaller radius r, you can write an expression for CD in terms of r and so write out Pythagoras for the triangle PCD.

This will give you a quadratic in r which you can solve.  I haven't done this yet myself but I expect one of the quadratic solutions is impossible for this problem.

EDIT:  The r squared terms cancel out so there's only one solution.

* I notice that those measurements don't work with my diagram.  Don't panic; just swap A and B over.

Bob

Last edited by Bob (2015-10-24 08:47:11)

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

math9maniac

Member

From: Tema

Registered: 2015-03-30

Posts: 443

Re: Circle problem

Thanks Bob for reply. Hope you're in good health. I'll return with feedback. Going out this morning. Thanks.

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Only a friend tells you your face is dirty.

math9maniac

Member

From: Tema

Registered: 2015-03-30

Posts: 443

Re: Circle problem

Thanks so much Bob. It worked. I'm truly grateful.

I have a question. The property of two circles having their centres on the same line, is it only true when the small circle touches the diameter of the larger? Will it always form a right-angle triangle? What if they touch externally? 

Thanks.

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Only a friend tells you your face is dirty.

math9maniac

Member

From: Tema

Registered: 2015-03-30

Posts: 443

Re: Circle problem

Please in another instance also can this problem be worked using a sort of ratio and proportion or general formula? Thanks.

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Only a friend tells you your face is dirty.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Circle problem

hi math9maniac

When we say two circles touch at a point, we mean that they share a common tangent at that point.  Otherwise they would intersect at more than one place.  The line from either centre to the point is at right angles to the tangent, so, as they share a point and a direction the two radius lines are the same line.

For the smaller circle to also touch a diameter you need to draw the a diameter with that property.  The diameter is then another tangent to the small circle, which is where the right angled triangle comes from.

If they touch externally, neither circle can touch the diameter of the other (unless you extend the diameter outside its circle).

Try this:

Draw a line CD and mark E anywhere on it between C and D.  Make a circle, centre C and radius CE, and a second circle, centre D and radius DE.  You will see that they touch at E.  Draw a line through E, perpendicular to CD.  This line is their common tangent.

Don't know about ratios and general formulas here.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

math9maniac

Member

From: Tema

Registered: 2015-03-30

Posts: 443

Re: Circle problem

Merci beaucoup. Je comprends maintenant.

Please I want a link on MIF for Matrices as well as Permutations and Combination. Also Mensuration on Solid Geometry. Thanks.

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Only a friend tells you your face is dirty.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Circle problem

hi math9maniac

"Je comprends maintenant. " Excellent.  Let me know when you have mastered "comprendre l'avenir". 

You will find matrices here:

http://www.mathsisfun.com/algebra/matri … ction.html

Perms and combs similarly.

Solid geometry is a huge topic.  There's a bit here:

http://www.mathsisfun.com/geometry/solid-geometry.html

but this doesn't cover calculations like angle between two lines or a line and a plane, etc.

If you have a specific question I'll try to do it and link it into my geometry contents page.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

math9maniac

Member

From: Tema

Registered: 2015-03-30

Posts: 443

Re: Circle problem

D'accord. Merci encore. Je t'aime Bob. S'il vous plaît, j'ai besoin d'un conseil. Il s'agit de mon avenir. "Members Only" sub-forum?

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Only a friend tells you your face is dirty.