As the number of sides in a polygon increases, the ratio of its perimeter to its longest diagonal approaches pi (3.14). Does this mean that for polygons with fewer sides, this ratio is less than 3.14? In the question, a 100-sided polygon is chosen over a 99-sided one, suggesting a larger ratio. So, is the ratio for a 99-sided polygon less than 3.14 compared to a 100-sided polygon?
Why doesn’t the question consider angle PCQ to be larger? If the line were tangent, angle PCQ would be 90 degrees. As the line gets closer to the angle PCQ, shouldn’t the angle increase in size? I’ve solved a similar problem before, and I believe my reasoning from that solution should accurately apply here as well.
Could you clarify the question below? I don’t understand why the area is smallest when the angle is 45 degrees. When I calculated it, the area at 45 degrees seemed largest compared to angles like 30 and 60 degrees (below and above 45 degrees). Could it be because a 45-degree angle forms a square, which has a larger area than a rectangle formed at other angles?
No? You just know that the ratio converges to \pi when n \to \infty, where n refers to the number of sides in the regular polygon of concern. This doesn’t justify whether Quantity B > Quantity A or the other way around.
In fact, Quantity A is greater than Quantity B.
Anyway, I did the math and if we define f as the function that denotes the ratio then we have:
f(2n + 1) = f(4n + 2)
f(2a) < f(2b) \iff a < b
Owing to those, you should have f(99) = f(198) > f(100).
Alternatively, if you do the math the regular way then you’re comparing between 198 \sin (\frac{\pi}{198}) and 100 \sin (\frac{\pi}{100}), which isn’t that nice to compare without a calculator. Anyway, I think this goes a bit beyond what u require for the GRE.
\angle PCQ can be 90 or whatever. Needless to say, “whatever” implies any valid angle in (0, 180).
Draw AC and EG (the diagonals of the respective squares) and note that EG > AC. What does that tell you about the side lengths of the corresponding squares? I think you can probably deal with the final bit of comparing the areas of those two squares.
Regarding question 1, yes, I am referring to a value less than pi. I am a bit confused; you mentioned that the ratio converges to pi as the number of sides increases, but this does not necessarily mean it becomes larger, correct? That was my original question, and I did not understand the mathematical explanation you provided (I would appreciate it if you could explain it again clearly). Since we are unsure whether it increases (although you mentioned that A is larger), is option D the correct answer for question 1?
Concerning question 2, does the concept of the tangent line not apply here? I was attempting to base my decision on it. Another question guided me to the correct answer using this decision-making process, so I am now somewhat confused.
Regarding the final question, I am unclear on the direction you mentioned. Could you please provide a more detailed and comprehensible explanation of the third question?
Consider the function above. Notice that y converges to 0 for increasing values of x. However, observing that y generally decreases toward 0 as x increases tells you nothing about specific values—such as f(4) versus f(7).
The same principle extends to your question at hand too. You know that for increasing side lengths, the ratio approaches \pi. This, however, doesn’t allow you to definitively conclude the relationship between f(100) and f(99).
We can’t say much just from the fact that the ratio converges to \pi for increasing number of sides. As such, we would have to do additional calculations to actually solve the problem and justify why A is indeed the right answer.
You don’t need to because it’s beyond what is required for the GRE.
It’s not that relevant. It’s just an isosceles triangle.
Is there something specific you don’t understand from the above hint?
So, regarding question 1, as you mentioned, since we just know about the convergence to the pi number as the number of sides increases, we should choose option D for question 1 , right? Because with this limited knowledge, we are unable to make a decision about which one is actually bigger! 99-sided or 100-sided.
In terms of question 2, I mistook it for the tangent line problem, since in that topic, we learned that we can draw a vertical line if the line is tangent to the circle, and the triangle formed would be a right-angled triangle. For this reason, I thought that if the line gets closer to the angle, it means the angle also becomes larger than 90 degrees. I am uncertain about the relationship between these concepts and find myself confused as to whether they can be connected at times. I am unable to determine the connection.
Regarding question 3, no, honestly:) Even I couldn’t understand why you stated EG is larger than AC! Perhaps by drawing a line in the 45-degree example, we can find some other values , but there is no way to do it with the 46-degree example. I would be thankful if you explained it to me. It might be straightforward, but I am struggling to find the solution or even follow your reasoning.
You can’t say anything JUST from the “pattern” (convergence), but you can definitely make a comparison if you do more work (like i briefly outlined above). When you do the extra work then you realize that A > B. On that note, was the answer provided for the question actually B?
That’s surely not a reason to choose D, is it? I think it’s fairly obvious that the answer is ofc not D, because you can imagine assigning this task to a computer and then actually computing the ratio using a drafting software or something of that sort—assuming you’re not familiar with the associated math.
Do we agree that A and C are midpoints of the outer square? Likewise, do you also see that E and G are points on the outer square that aren’t midpoints?
