Geometry Tangent Help

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OX can be seen to be [MATH]\sqrt{r^2-64}[/MATH] from the right-angled triangle XOY (Write it on your diagram.)
MX is x+5
Find an expression for the length MO using the right-angled triangle MXO
Find an expression for the length MO using the right-angled triangle MNO
Equate these two expressions and solve for x.
 
Beer induced reflection follows.
pka said:
Spersaw100 said:
View attachment 27627
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You will need the tangent-secant theorem.
\((ML)(MK)=(MN)^2\) the length of the tangent is a mean proportional between the lengths of the external part and the whole secant.
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`overline(NM)` is indeed easily solved with the tangent-secant theorem but that diagram by lex at post #2 made me wonder if it's possible to also solve the radius of the circle with the given data. Exploring a few ideas led me in circles so far. Need more beer. Maybe a few shots of brandy might be in order.
 
jonah2.0 said:
`overline(NM)` is indeed easily solved with the tangent-secant theorem but that diagram by lex at post #2 made me wonder if it's possible to also solve the radius of the circle with the given data. Exploring a few ideas led me in circles so far.
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Yes, it turns out that r can be anything greater than 8. You can't solve for it, no matter how much you drink.

And that should be obvious, if you think about it enough (though it wasn't to me at first). The tangent-secant theorem tells you that, for any circle, the indicated configuration has the same lengths. Once you've solved for x, if you get ML and MN the right lengths, the 16 follows.
 
jonah2.0 said:
Beer induced reflection follows.

¯¯¯¯¯¯¯¯NMNM¯ is indeed easily solved with the tangent-secant theorem but that diagram by lex at post #2 made me wonder if it's possible to also solve the radius of the circle with the given data. Exploring a few ideas led me in circles so far. Need more beer. Maybe a few shots of brandy might be in order.
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You are going around in an uncountably infinite number of circles. The context obviously requires r to be greater than 8. As long as r is greater than 8, my solution shows that x is independent of r and can be found no matter what the value of r (>8).
Hopefully you can enjoy the brandy, all the more!
 
Beer and brandy induced disappointment follows.
Dr.Peterson said:
Yes, it turns out that r can be anything greater than 8. You can't solve for it, no matter how much you drink.

And that should be obvious, if you think about it enough (though it wasn't to me at first). The tangent-secant theorem tells you that, for any circle, the indicated configuration has the same lengths. Once you've solved for x, if you get ML and MN the right lengths, the 16 follows.
Click to expand...
I had a feeling I was chasing after my own tail especially after doing a simulation with a circle with a known radius and imposing similar conditions. A Google search of "find radius of a circle from a tangent and a secant"
https://www.google.com/search?q=fin...ZgBAKABAcgBCMABAQ&sclient=mobile-gws-wiz-serp gave some ideas but nothing useful. I was almost sure it could be done. Thanks for the opinion.
lex said:
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Hopefully you can enjoy the brandy, all the more!
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I usually do but I guess I should have gone for the tequila this time; I wouldn't have minded exploring more ideas for hours happily even if I knew in my heart that it's useless to go on. Kinda like doing something unpleasant but you just don't care because you're happily inebriated. Reminds me of that movie starring Leonardo DiCaprio when he asked his business partner for coke as he was facing what seemed like certain death. He just didn't want to die sober.
 
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