Brain Teasers
Circle's Corner
Consider the two lines AO and BO. These two lines intersect at point O and are perpendicular to each other.
If you were to draw a circle that were to intersect these two lines at points A and B and only at points A and B, what would be the smallest integral size of the larger arc AB in degrees?
If you were to draw a circle that were to intersect these two lines at points A and B and only at points A and B, what would be the smallest integral size of the larger arc AB in degrees?
Hint
Integral means "in integers".An arc is a portion of a circle.
Answer
181 degrees.If you created a circle with points A, B, and O on it, since angle AOB is a right angle, line AB is the diameter of this circle. Therefore, arc AB would be 180 degrees if it actually were to intersect O as well as A and B.
However, the problem says the arc can only intersect A and B. To do this, the circle must be slightly larger than the one described above. As stated in the problem, the larger arc must be the next largest integer size, which is 181 degrees.
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Comments
If i understand the question completely and correctly, then i disagree with the answer. I think it should be 179. I agree that at 180 degrees, point O would have to fall on the circle. But in order to reach an arc of 181, then O would have to fall outside of the circle, causing the circle to intersect AO and BO at points other than A and B, which violates the stated condtions. O would have to fall inside the circle, which would create an arc less than 180. Next integral value would be 179 degrees.
I believe the answer and wolfpack's correction are both wrong. If AO and BO are lines and the circle only intersects with these lines at A and B respectively, then they must be tangent lines at A and B. So O could not be on the circle and would in fact be outside the circle. Let's call the center of the circle X. AOB would be 90 degrees because AO and BO are perpendicular and both XAO and XBO would also be 90 degrees because AO and BO are tangent lines. Therefore, the central angle AXB would also be 90 degrees and the major arc AB could only be 270 degrees.
Jota, I completely agree with you. I saw this teaser in review and I'm surprised it made it onto the site. I just submitted a correction to this teaser, so we'll see what the Editors think of it.
I am stupid and I barely got past the first sentence.
I saw this a few days ago as well and agree that it is wrong. More directly, perhaps -
(a) it was explicitly stated that intersection was ONLY at A and B (and thus NOT at O);
(b) the construct cited "lines" twice - not line segments as the answer implies. "Lines" are infinite in both directions, and thus for a circle to intersect "only at A" and "only at B" there is no other possibility other than tangency - at both points. (In other words, if not tangent, then the circle would have to intersect elsewhere, too - maybe on the far side of O - but this is ruled out clearly by the statement).
If tangent to two points, and the tangency lines are perpendicular, there is no possibility other than the two points being 90 degrees apart on this circle. (Equivalently, as a previous poster correctly stated, "270 degrees".)
If something else was intended, the problem is ill-stated.
(a) it was explicitly stated that intersection was ONLY at A and B (and thus NOT at O);
(b) the construct cited "lines" twice - not line segments as the answer implies. "Lines" are infinite in both directions, and thus for a circle to intersect "only at A" and "only at B" there is no other possibility other than tangency - at both points. (In other words, if not tangent, then the circle would have to intersect elsewhere, too - maybe on the far side of O - but this is ruled out clearly by the statement).
If tangent to two points, and the tangency lines are perpendicular, there is no possibility other than the two points being 90 degrees apart on this circle. (Equivalently, as a previous poster correctly stated, "270 degrees".)
If something else was intended, the problem is ill-stated.
How many of these midpoints can any correction avoid? AOB is a right triangle. AB is the hypothenuse. For the smallest possible circle, AB = D, i.e. the chord AB is a diameter. O must also be a point on that circle. If AB is any other chord, it will be from a larger circle, and either the circle or its mirror will inclose O. In stead of a 'tease,' we get an 'insipidity.' If a circle is divided into two non-equal arcs the smallest the larger can be is just over 180 degrees.
Stil, the problem statement explicitly rules OUT the case of point O lying on the circle. To quote directly:
"...circle that were to intersect these two lines at points A and B and only at points A and B..."
O is not a point of intersection. A and B are the only points of intersection. There is no other construct consistent with this other than the 90 degree (or 270 degree) arc.
"...circle that were to intersect these two lines at points A and B and only at points A and B..."
O is not a point of intersection. A and B are the only points of intersection. There is no other construct consistent with this other than the 90 degree (or 270 degree) arc.
If the two lines AO and BO are actually only line segments and point O is inside the circle, then the given answer (181) is correct. If point O is outside the circle, then 270 is bigger than 181.
But since the problem said two lines, then 270 is the answer.
But since the problem said two lines, then 270 is the answer.
Yep. 270 is the only answer, not just the smallest, largest, or integral.
I figured the rest of the information was just being thrown in as a distraction, but I guess it distracted cogmatic.
I figured the rest of the information was just being thrown in as a distraction, but I guess it distracted cogmatic.
Yeah. It's impossible to have any other angle other than 180 (intersect A, B, and O) or 270 (intersect A and B). Any other angle would require the lines AO and BO to NOT be at right angles.
And if it's integer use integer -_-" I nearly thought he was asking for integration. For goodness sake, with that little infomation, no way am I going to kill my brain cells finding the area bounded by the arc! (Possible if you let x/y be the length of AO/BO, find the flattest arc length, form an equation, integrate, etc...)
And he still dares to send an offensive filled pm to swear at and insult me for commenting on one of his other equally bad teaser.
And if it's integer use integer -_-" I nearly thought he was asking for integration. For goodness sake, with that little infomation, no way am I going to kill my brain cells finding the area bounded by the arc! (Possible if you let x/y be the length of AO/BO, find the flattest arc length, form an equation, integrate, etc...)
And he still dares to send an offensive filled pm to swear at and insult me for commenting on one of his other equally bad teaser.
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