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## #1 2014-02-02 22:34:02

thedarktiger
Member
Registered: 2014-01-10
Posts: 91

### Triangles and tangents galore

Triangle ABC, inscribed in a circle, has AB = 15 and BC = 25. A tangent to the circle is drawn at B, and a line through A parallel to this tangent intersects \overline{BC} at D. Find DC.
This is weird. I tried stuff I know 'bout circles and triangles...(which is pretty close to nil)...but I can't get it.
Thanks!
:D:D

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## #2 2014-02-02 23:23:05

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

### Re: Triangles and tangents galore

Hi thedarktiger;

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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## #3 2014-02-03 21:17:46

bob bundy
Registered: 2010-06-20
Posts: 8,016

### Re: Triangles and tangents galore

Good morning bobbym,

Did you get that answer from geogebra or have you found a 'Euclidean' method.

I'm finding a solution to this one elusive at the moment.

Bob

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

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## #4 2014-02-03 21:24:42

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

### Re: Triangles and tangents galore

Hi Bob;

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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## #5 2014-02-03 23:22:54

bob bundy
Registered: 2010-06-20
Posts: 8,016

### Re: Triangles and tangents galore

It's a strange problem.  There are an infinite number of triangles with those measurements.  I used Sketchpad to fix AB and then made a circle, centre B, for the locus of C.  Then I constructed the rest to find D.  As I move C, D moves around a circle of fixed radius, with a radius consistent with your answer for CD,  and centre B.  But why?

I seem to have landed up with two problems.  (i) Why is the locus for D a circle ? and (ii) what is its radius because once you know BD, you know DC.

??????

EDIT:  Oh how come I failed to see this earlier:

(cannot add a diagram to an edited post so I'll put in the next post )

angles EBA BCA and DAB are equal (angle props and parallels)

so

triangles DAB and ACB are similar

so AB/BD = CB/AB

I'll leave the rest to you.

Bob

ps.  But why the red circle ?  working on that now.

No I'm not.  I've done it.  BD is fixed so D moves on a circle with that radius.

Last edited by bob bundy (2014-02-04 01:20:33)

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

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## #6 2014-02-04 01:16:32

bob bundy
Registered: 2010-06-20
Posts: 8,016

### Re: Triangles and tangents galore

diagram

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

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## #7 2014-02-04 04:19:33

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,592

### Re: Triangles and tangents galore

Here is a little video capture of my Geogebra program showing that DC's size remains constant at 16 when BC is pivoted 360° around point B.

Last edited by phrontister (2014-02-04 17:29:40)

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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## #8 2014-02-04 05:03:14

bob bundy
Registered: 2010-06-20
Posts: 8,016

### Re: Triangles and tangents galore

Thanks Phro, that's lovely.  Like the music too.

Bob

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

Online

## #9 2014-02-04 07:50:47

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

### Re: Triangles and tangents galore

Hi phrontister;

Nice video.

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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## #10 2014-02-04 17:49:58

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,592

### Re: Triangles and tangents galore

Thanks, B&B!

I've changed the link in that post to a longer version that oscillates, so that Bob can hear the whole song...which is The Engagement, apparently from an old movie, Silent Partner.

Just make sure that you're well-rested before viewing this one, or the hypnotic effect might getcha.

Watching the video, I imagine a huge albatross in flight. Maybe I could've used Fleetwood Mac's Albatross for the audio instead.

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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## #11 2014-02-04 19:53:37

bob bundy
Registered: 2010-06-20
Posts: 8,016

### Re: Triangles and tangents galore

Thanks Phro,

That was very relaxing.  Yes, Fleetwood Mac would work well but, then again, I've got that one on vinyl so it was good to hear something new.

Bob

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

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## #12 2014-02-07 14:54:50

thedarktiger
Member
Registered: 2014-01-10
Posts: 91

### Re: Triangles and tangents galore

Hmmm. Thanks for the demonstration phronstister! wow, It looks like geoebra is  amust

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## #13 2014-02-07 15:23:27

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

### Re: Triangles and tangents galore

Some old guy wrote:

Geogebra. geogebra, it is the man. If it can not do it, then nobody can! Yaaaaay, geogebra!

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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## #14 2014-02-07 23:40:31

bob bundy
Registered: 2010-06-20
Posts: 8,016

### Re: Triangles and tangents galore

hi bobbym,

Can I make a claim for the first mention of geogebra on MIF ?

http://www.mathisfunforum.com/viewtopic … 01#p146001

Bob

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

Online

## #15 2014-02-08 00:05:30

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

### Re: Triangles and tangents galore

Hi;

I would have to agree.

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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## #16 2014-02-08 00:06:09

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,592

### Re: Triangles and tangents galore

And I'll make a claim for the most recent (to date) mention of geogebra on MIF.

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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## #17 2014-02-08 00:07:19

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

### Re: Triangles and tangents galore

The Alpha and the Omega!

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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## #18 2014-02-08 16:30:35

thedarktiger
Member
Registered: 2014-01-10
Posts: 91

so it is

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## #19 2016-07-31 12:45:31

quantumpizza21
Member
Registered: 2016-07-31
Posts: 2

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## #20 2016-07-31 13:22:37

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

### Re: Triangles and tangents galore

Hi;

Geogebra allows you to experiment. It allows you to spot patterns and relationships that you would never have seen without it. It combines the best aspects of math and programming into a simple interface.

When I started learning geometry we were warned by our teachers not to pay any attention to the diagram that came with the problem. Do not try to think that AC might equal AB based on my diagram, they would say. Well, back then when we wrote on animal hide scrolls and used quill pens the drawings were so bad that they were right. But supposing drawings of geometric figures were made to high accuracy, with scale all accurate, what then? Turns out, the human eye is very good in estimating distances and angles. We were doing that way before anyone ever heard of Euclid.

Because geogebra can measure and draw very accurately it allows the problem solver to use his eyes as well as his brain. Relationships pop out very quickly and a whole new method of solving problems is the result. I daresay that if every schoolkid in the world used geogebra they would not need forums like this one for help with their schoolwork because fully 90% of the problems brought in here can be done using geogebra in a couple of minutes.

In addition, it can assist in the learning of geometry as taught in schools, you know, the old fashioned way. Unbelievable to see people that are 70 or 80 years younger than I being so old fashioned. But if you must learn geometry the same way Euclid thought about it then geogebra can help there too. The whole concept is lumped into the new old concept called Experimental Math, EM for brevity. Welcome to the forum.

Check out some neat uses of the gebra in the Computer Math thread. Try them out for yourself.

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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