Geometry Tangents Help

J

JacobH

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Apr 18, 2020
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I haven't done any of these in months and need a refresher to get back on my feet. Find "X".
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Frankly, these are so easy it's hard to give a hint without giving the answer!

But you need to know one fact that whoever gave you these problems clearly expects you to know: If an angle, with vertex outside a circle, cuts off two arcs of the circle, then the measure of the angle is the arithmetic average of the measures of the arcs. I'm sure that's in your text book- probably very close to this problem set!
 
I think this first one needs some extra information to solve it...
1587262165250-png.18004


Is it perhaps given that line AC is a tangent to the circle (so angle PAC is a right angle) ?

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The rest of them seem solvable with the method highlighted by @HallsofIvy , although he made a slight mistake in quoting it. Rather than "arithmetic average" it's "arithmetic difference divided by two". See this page for details and a nice proof (click).
 
There is no way to get the that answer - with the given information.

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pka said:
Sorry Prof Klan but that in incorrect.
\(\|\overline{PB})\|=\|\overline{PA})\|=4\) so that \(\|\overline{PC}\|=12\)
Hence \(x=8\sqrt{2}\)
Click to expand...

That's obviously the intended answer then - because \( 8\sqrt{2} = \sqrt{128}\) which agrees with post #4

However the question didn't state that angle PAC is a right angle therefore Subhotosh and I are correct that, as stated, it couldn't be solved. No time in the (90o?) corner for us :)
 
Cubist said:
However the question didn't state that angle PAC is a right angle therefore Subhotosh and I are correct that, as stated, it couldn't be solved. No time in the (90o?) corner for us
Click to expand...
YES IT DID. Did you both bother to read the title of the thread?
The title is Geometry Tangents Help.
So why would either of you think that tangents are not implied in all of the given questions?
Please tell me! I have edited for testing for years, therefore I think that I understand the language of testing.\
This may be an exception to that norm, but I would be greatly surprised if it were.
 
pka said:
YES IT DID.
Click to expand...

No it didn't - because it wasn't stated that A is the point of tangency nor that P is the circle centre. But I'm just playing devil's advocate now ? ?

Being honest with you I did miss the title ?

JacobH - have you managed to solve the other ones now?
 
pka said:
YES IT DID. Did you both bother to read the title of the thread?
The title is Geometry Tangents Help.
So why would either of you think that tangents are not implied in all of the given questions?
Please tell me! I have edited for testing for years, therefore I think that I understand the language of testing.\
This may be an exception to that norm, but I would be greatly surprised if it were.
Click to expand...
Because one of the problems:

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I would not know where the tangent/s are hiding?

@pka, Did you bother to analyze the whole problem set?
 
Subhotosh Khan said:
I would not know where the tangent/s are hiding?
Click to expand...

You simply need to extend the vertical line of the "1" in the "140" downwards to obtain the tangent. Please don't be too fussy :ROFLMAO:

On a serious note the drawings above annoyed me when I first saw them. I was always drilled that my sketches should be clear and explicitly show right angle symbols, equal length lines, arcs to enclose an indicated angle etc. In an exam (a timed) situation it is sometimes useful to produce a sketch. And without using these various notations it can be hard to quickly produce a sketch where all the appropriate lengths look different and angles that aren't 90 actually look far away from 90, etc. Assumption can easily lead to a mistake

BTW: @JacobH please don't let this discussion put you off asking more about your question if you are still stuck.
 
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