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In right triangle, ABC, the length of side AC  is 8  the length of side BC  is 4  and C is 90 degrees The circumcircle of triangle ABC is drawn. The angle bisector of ACB meets the circumcircle at point M  Find the length CM

Oct 2, 2021

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deleted as in error.

Oct 3, 2021
edited by Melody  Oct 3, 2021
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CM is NOT diameter!!!

CM would be the diameter only if AC = BC!!! But this is not the case.

Hint:     arc AM = arc BM

Guest Oct 3, 2021
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You are right guest, CM is definitely not a diameter. I apologize for my error.

Melody  Oct 3, 2021
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I can see from my diagram that MB=AM   but the reason eludes me....

How did you determine this?

Melody  Oct 3, 2021
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A while ago I've seen Phill's graph.

So, CM happens to be √32   (2√8)

Neat job, Melody!

Guest Oct 4, 2021
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CM = √72    or   3√8

I guess... nobody's perfect.

Guest Oct 4, 2021
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Thanks

How did you determine this?

Melody  Oct 4, 2021
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Here's Phill's graph  (https://web2.0calc.com/questions/help_37192)

If AM = BM then these 2 arcs have the same length.

AM = √50        BM = √50

Oct 4, 2021
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If AM = BM then these 2 arcs have the same length.

Yes that is obvious, but you have not said why  AM=BM    That is what I keep asking you.

Thanks for including the address of Chris's explanation. Now i understand.

The explanation is that since AB is a  diameter, the radius and the centre of the circle are easy to determine.

From that information, all else can be determined.

Oct 5, 2021
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I think the main thing that causes this is the line segment CM being an angle bisector of the right angle ACB.

Angle AMB must be 90 degrees!

No matter how much you increase/decrease AC and BC, AM = BM !!!

To me, this is the phenomenon.

There must be some logical explanation.

Guest Oct 5, 2021
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oh yes there is a logical reason, .

Any angle in a semicircle will be 90 degrees.

This is an extension of the  theorem that:

Any angle on circumference subtended from an arc will always be half that as will be subtended to the centre.

(my wording may leave a little to be desired)

I knew AB was a diameter but I did not think to use the centre in my logic.

Melody  Oct 5, 2021
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Maybe I misunderstood your delema?  It doesn't matter though.  Time to move on.  :)

Melody  Oct 5, 2021