Angle ABC is 40 degrees and the area of the circle is 81...

[NileshG679]

Angle ABC is 40 degrees and the area of the circle is 81pi. If CB is a diameter of the circle, how long is the arc AXC?

This is from MGMAT Prep Guide 4, Problem set 7, #15.

My question is related to central and inscribed angles. If a side of the inscribed angle goes through the center of a circle, can you still have a central angle? I would assume that the central angle would be drawn differently than the explanation given on page 77 of the same guide. With regards to the question, if the inscribed angle is 40 degrees, and since one of the sides of the angle goes through the center of the circle, will the central angle = 80 degrees?

Thanks!

[RonPurewal]

the measure of ANY inscribed angle is 1/2 of the number of degrees in the arc (= 1/2 of the measure of the corresponding central angle).

if the arc is less than 180º, then it's possible for an inscribed angle to pass through the center of the circle.
whether it actually does so, however, is irrelevant-- the result is the same either way.

[MD91]

For this particular problem, I took a different approach and got a slightly different answer. I drew a straight line connecting A and C line and that way got an inscribed triangle ABC. Now, since BC is a diameter of the circle it follows that the angle across it (CAB) has to be a right angle (90 degrees). We also know that ACB has to be 50 degrees since 180- 90 (right angle CAB) - 40 (given angle ABC) = 50. So, I drew another straight line from point A to the center of the circle (let's say, point O). This way I got a new small triangle ACO. The angle AOC in that triangle is what we actually need to determine the length of arc AXC. Now,we know that the sum of all angles in the small triangle ACO has to be 180. We can conclude that angle OCA will be equal to BCA (literally it is the same angle), which is 50 degrees. Then this next step is crucial: the way I found angle CAO - I concluded that CAO has to be equal to half of the right angle CAB because line AO bisects the line (CB=diameter) that is across the angle, and thus it is also bisects the angle CAB. Thus, CAO is half of CAB = 45. Now, we know that CAO+ACB+AOC=180 which follows that AOC = 180 - 45 -50 = 85. This angle is what I am getting different from the official answer, which leads me to a slightly different final answer. If someone can point me out where I am making a mistake I would greatly appreciate it!! Thanks!

[RonPurewal]

you're bringing way too much artillery to this fight...
your triangle ACO is isosceles, since each of AO and CO is a radius of the circle. so, angle AOC = angle ACO = 50º.

[RonPurewal]

incidentally, here is where you went astray:

I concluded that CAO has to be equal to half of the right angle CAB because line AO bisects the line (CB=diameter) that is across the angle, and thus it is also bisects the angle CAB

...nope. this is not a thing.

to see why, just draw a triangle with angles like 10º-80º-90º (instead of 40º-50º-90º), and draw the same line to the center of the circle. then, you'll still have two equal radii, but it will be very obvious that the right angle is not bisected.

[MD91]

Thanks for the reply and clarification. I guess I was complicating the thing too much and managed to make up my own theorem that is not true (the one about bisecting lines and angles).

[tim]

Glad to see you learned something from this one. Let us know if you have any more questions.

[RonPurewal]

there's nothing wrong with making up conjectures.
in fact, if you're making up novel conjectures, that's a very good sign: it means that you're actually thinking creatively about the situation at hand, rather than just trying to memorize a bunch of stuff.

that kind of thing—'thinking on your feet', or whatever else you want to call it—is arguably the single most important aspect of a successful GMAT quant mentality.

also, it's a leadership quality. leaders innovate. followers memorize.

--

(nerd corner:
it's not a 'theorem' unless you have actually [i]proved[li] it. until then it's just a 'conjecture', which is just a smart-kid word for 'guess'.)

[RonPurewal]

what's missing here, though, is the kind of thinking that should come after 'creative' thinking... namely, critical thinking. if you come up with a new conjecture, TEST IT.
test the living heck out of it. take every different kind of case you can imagine to which your conjecture applies, and throw all of them at it. if the conjecture survives, then it's probably a valid principle.

here, your conjecture would not have stood up to simple testing (with right triangles that don't look like 45-45-90). but that's all part of the process—you'll make up a lot of things that don't actually work.

remember—proving that something doesn't work is just as significant as proving that it does!
in other words, 'X IS NOT a rule / X has exceptions' adds to your overall knowledge (and intuition) every bit as much as 'X IS a rule' does.

[tim]

Great advice, Ron. Probably more important than anything we could ever teach about the GMAT proper. I wish more of our students would take the time to listen to us when we tell them how much damage they're doing to their GMAT chances (and future careers) when they try to reduce common sense, creativity, and insight down to something they must memorize.

[RonPurewal]

the origin of that issue ^^ is that people are (VERY mistakenly) associating the GMAT exam with 'school tests', on which memorization is, in fact, useful.

the mistake lies in thinking that the GMAT will work like 'school tests'.
the reality is EXACTLY the opposite: THE ONLY REASON THE GMAT EVEN EXISTS is to be COMPLETELY DIFFERENT from 'school tests'.

everyone taking this exam has at least 16 years of school transcripts. therefore, if the GMAT required the same skill set as school tests, it would add no extra value, and would thus have no reason to exist.