I’m an engineer, so I do know my way around simple Math, and I’d certainly expect to ace middle-school level algebra. Then my son came home with this problem and I’m stumped. I think there’s some missing data. So, if you can, please help. Am I rusty? Overlooking something? An idiot?
Regards,
Edit: Forgot to add; he’s supposed to find x. There isn’t any more information in the question, just “Find x” and the image.
There’s a geometry theorem that if you take 2 points on the circumference of a circle, the angle formed between those 2 points and the center is exactly twice the angle between those 2 points and any other point on the major arc of the circle. I don’t remember the name of it. So angle BAC is 30 degrees, and it’s just algebra from there.
And I only know this because my daughter brought home a similar geometry problem earlier this year.
Too late to edit:
Here’s a proof.
5x+20=30
x=2
It is just combining geometry with algebra.
Although it doesn’t help that the black dot for the 60 degree angle seems to be off-center.
As another engineer that missed something similar on the GRE, I sympathize.
“O” is the standard symbol for a center of a circle. At least in my daughter’s textbook.
Is it intended from the picture that O is the center of the circle, and that a line drawn through AO would bisect the angle BOC?
If so, then:
Angle AOB is 150 degrees (because it’s the supplement to half the angle BOC).
Triangle AOB is isoceles (since AO and OB are radii of the circle).
And thus, BAO is 15 (to make the triangle AOB’s angles add to 180).
In which case you have 5x + 20 = 2*15, and thus x = 2.
Edited to add: I guess it’s probably not required that the line drawn through AO would bisect BOC, since the result should hold either way by the Central Angle Theorem, as muldoonthief says.
Wow, thanks a lot for that (and so fast!). I may delay teenage derision for a few more days. Thank you!
Slight aside, but I think with these problems they should just tell you the answer, and then get you to explain why (or only give full marks if full working is included, but that needs to be made explicit upfront).
Because otherwise they are easy to guess. There must be a relationship between these angles for this problem to be solvable, and the angles visually look twice the size. Done.
Agreed
x=2
Crane
Doesn’t telling someone the answer just CONFIRM the relationship that they’re already guessing at? Or at least allow someone to rule out other possibilities they might consider (like that the angles are congruent, or sum to 180°, etc.)?
I see now I could extend the reasoning I used in post 7 to the case where the line through AO doesn’t bisect BOC.
Basically, you draw the line anyway, call the portion of angle BOC that lies above the line X, and call the portion of BOC that lies below the line Y.
From the reasoning above, angle AOB is the supplement to X, and the other two angles of triangle AOB are equal. Similarly, AOC is the supplement to Y, and the other two angles of triangle AOC are equal. Again, we also know that the angles of any triangle add to 180.
This gives:
2BAO + (180 - X) = 180
2CAO + (180 - Y) = 180
Which implies BAO = X / 2 and CAO = Y / 2.
And thus since BAC = BAO + CAO, we have BAC = (X + Y) / 2
My intent here was to show that it’s possible to reason it out without knowing the central angle theorem. But in doing so, it turns out I’ve basically sketched a proof of the central angle theorem. 
I solved it in a couple of minutes without knowing this theorem but using simple trigonometry.
Assume that the radius of the circle is 1.
Let D be the middle of the segment BC. Then tan(DAB)=BD/AD.
BD = sin(30)
AD = AO+OD = 1+cos(30)
It take my calculator and find DAB=15.
So BAD=30, etc.
As an aside, am I the only one who gets really annoyed by problems that “combine algebra and geometry” in this way? They’re totally artificial: You’ll never actually encounter a problem like this outside of a high-school textbook. 5x+20 = 30? Sure, that’s a reasonable problem to encounter. Central angle theorem? That’s useful. But why would the angle in your central angle theorem ever be (5x+20)º?
At least this one is consistent about the units, though. Too often, I see them with an angle being given the same units as a side, or with an angle of “x^2 -2x +1”, or the like.
Of course it’s artificial. The vast majority of problems in math textbooks and tests are. But how is it any more artificial than 5x + 20 = 30 without any additional context?
The fact that the problem puts together things that don’t normally go together is a feature, not a bug. It means that solving the problem involves at least a little bit more than mindlessly following the procedure for solving a problem of a certain type.
Not to mention the fact that the 60 degree angle looks like a right angle.
So I brought up a screen grab of this in my CAD software and the circle is not even a circle. It is too tall. So I squished the height by around 3% and it lines up pretty well with a circle.
And you’re right - the true center of the circle (in the fixed image) is a few pixles to the left and a few pixels down. It’s off by about the radius of the black dot at an angle of about 45 degrees down and to the left.
And the 60 degree angle measures about 83 degrees. And the other linework doesn’t match up either - C is not directly under B and A is too high on the circle.
It’s a pretty poorly drawn image.
Yeah - I thought that was weird too. It’s a geometry (or trigonometry) problem, with the 5x+20 thing thrown in for no apparent good reason. If you can figure out the angle, the algebra is simplistic at best.
The only thing I can think is it reinforces the student to carefully consider when the problem is totally done. Sometimes when you think you’re done (in a good problem) there’s one final step to get the “answer”.
For example, Jack and Jill have 12 nickels between them and Jill has twice as many nickels as Jack does. How much money does Jack have?
Some students might stop at figuring out the number of nickels each has and forget to figure out how much money that equals for Jack.
But out of any context, the algebra in this problem just seems like an awkward add-on.
That annoyed me as well. I tried to think of a parody that captures what it is about it that bugs me, and I came up with this:
Just because you can glue two unrelated problems together, doesn’t mean you should.
It’s not that hard to come up with a wide variety of contexts for that algebra problem. A football player starts off on his own 20 yard line. He runs at 5 yards per second downfield. At what time will he cross the 30 yard line? A cheap motel charges $20 a night for a room, plus $5 per person. How many people can stay in the room for $30? Problems with the same general form as that show up all sorts of places. But can you think of any plausible place the OP’s problem would come up?
And yes, a lot of other textbook problems are also highly artificial. Those annoy me, too. There are enough non-artificial problems in the world to fill the books.
Because in real life, you may very well encounter a real-world problem that reduces down to 5x+20=30. You may also encounter a situation where you need to know this particular angle. But this particular combination of problems is completely unrealistic.
Sure, but this particular problem doesn’t feel like a useful way to test a student’s ability to adapt. It’s some lazy teacher’s idea of a problem that combines multiple skills.
