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805+ Level|   Geometry|               
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Braintree
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A pentagon with 5 sides of equal length and 5 interior angles of equal Updated on: 27 Sep 2022, 12:52
Originally posted by Braintree on 22 Sep 2022, 23:33.
Last edited by Braintree on 27 Sep 2022, 12:52, edited 1 time in total.
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Consolidating my approach for this question:

For any regular polygon, the limit of the ratio of the perimeter and the longest possible line segment is pi (which is when the number of sides approach infinity- i.e. a circle). The diagonal is the longest line possible in a regular pentagon. So the ratio of this diagonal to the perimeter of the regular pentagon has to be less than pie. Therefore, if the diagonal of this pentagon is less than 8, its perimeter has to be less than 26.

On why the diagonal would be the longest line inscribed in a regular pentagon: If we eyeball (or apply the pythagoras theorem to) the isoscles triangle formed by two of the diagonals and one of the sides of the pentagon, we can see that the sides of that isoscles triangle will always be greater than any other line drawn from the vertex of that triangle to the base of the triangle. Therefore, we can infer that the diagonal would indeed be the longest line possible in a regular pentagon.

ps. Just trying to connect the dots between the GMAT question maker’s mindset, and the enigmatic pi.
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A pentagon with 5 sides of equal length and 5 interior angles of equal 27 Sep 2022, 07:44
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Very well said, Braintree! The only tweak I'd make is here:
Braintree
For any regular polygon, the limit betweenof the ratio of the perimeter to the longest straight line possible segment is pi
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A pentagon with 5 sides of equal length and 5 interior angles of equal 16 Oct 2022, 04:55
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We can also use the property that the perimeter of a polygon inscribed in a circle can never be more than the circumference of the circle and the fact that the diagonal will be less than 2r.
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A pentagon with 5 sides of equal length and 5 interior angles of equal 20 Nov 2022, 02:42
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ScottTargetTestPrep
Bunuel
A pentagon with 5 sides of equal length and 5 interior angles of equal measure is inscribed in a circle. Is the perimeter of the pentagon greater than 26 centimeters?

(1) The area of the circle is 16π square centimeters.
(2) The length of each diagonal of the pentagon is less than 8 centimeters.



DS75271.01
OG2020 NEW QUESTION

Solution:

We are given a regular pentagon inscribed in a circle. We need to determine whether the perimeter is greater than 26 cm. If it is, then each side has to be greater than 26/5 = 5.2 cm. We need to know the following fact:

If a regular pentagon with side length s is inscribed in a circle of radius r, then:

s ≈ 1.18r

Statement One Only:

The area of the circle is 16π square centimeters.

We see that the radius of the circle is √(16π/π) = 4 cm. So a side of the inscribed regular pentagon is approx. 4(1.18) = 4.72 cm, which is less than 5.2 cm. Therefore, the perimeter of the pentagon is not greater than 26 cm. Statement one alone is sufficient.

Statement Two Only:

The length of each diagonal of the pentagon is less than 8 centimeters.

Each diagonal of a regular pentagon has the same length, and we need to use the following fact:

If a regular pentagon has a side length of s and a diagonal of length d, then:

d ≈ 1.62s

Since each diagonal of the pentagon is less than 8 cm, each side of the pentagon is less than 8/1.62 ≈ 4.94 cm, which is less than 5.2 cm. Therefore, the perimeter of the pentagon is not greater than 26 cm. Statement two alone is sufficient.

Answer: D
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Hi Scott ScottTargetTestPrep, could you elaborate a bit more on how did you derive 1.18r & 1.62s please? Thanks Scott
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A pentagon with 5 sides of equal length and 5 interior angles of equal 23 Nov 2022, 16:07
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ScottTargetTestPrep
Bunuel
A pentagon with 5 sides of equal length and 5 interior angles of equal measure is inscribed in a circle. Is the perimeter of the pentagon greater than 26 centimeters?

(1) The area of the circle is 16π square centimeters.
(2) The length of each diagonal of the pentagon is less than 8 centimeters.



DS75271.01
OG2020 NEW QUESTION

Solution:

We are given a regular pentagon inscribed in a circle. We need to determine whether the perimeter is greater than 26 cm. If it is, then each side has to be greater than 26/5 = 5.2 cm. We need to know the following fact:

If a regular pentagon with side length s is inscribed in a circle of radius r, then:

s ≈ 1.18r

Statement One Only:

The area of the circle is 16π square centimeters.

We see that the radius of the circle is √(16π/π) = 4 cm. So a side of the inscribed regular pentagon is approx. 4(1.18) = 4.72 cm, which is less than 5.2 cm. Therefore, the perimeter of the pentagon is not greater than 26 cm. Statement one alone is sufficient.

Statement Two Only:

The length of each diagonal of the pentagon is less than 8 centimeters.

Each diagonal of a regular pentagon has the same length, and we need to use the following fact:

If a regular pentagon has a side length of s and a diagonal of length d, then:

d ≈ 1.62s

Since each diagonal of the pentagon is less than 8 cm, each side of the pentagon is less than 8/1.62 ≈ 4.94 cm, which is less than 5.2 cm. Therefore, the perimeter of the pentagon is not greater than 26 cm. Statement two alone is sufficient.

Answer: D

Hi Scott ScottTargetTestPrep, could you elaborate a bit more on how did you derive 1.18r & 1.62s please? Thanks Scott
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Both of those numbers are obtained using trigonometry. If you have a basic understanding of trigonometry, you can refer to this page for the first number and this YouTube video for the second number.
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A pentagon with 5 sides of equal length and 5 interior angles of equal 21 Apr 2023, 07:08
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Tusharminj
My approach for Solution for Statement 2:

Given: Diagonal of pentagon is less than 8 centimetres. We can safely say that the diagonal of the circle will be greater than the diagonal of pentagon. Let's take the smallest such possible value for diagonal of circle, ie. 8 cm. Hence the perimeter of circle is 8x3.14 = 25 ish. Since pentagon is inscribed inside the circle, its perimeter will be lesser than that of the circle. Hence the perimeter of the pentagon will be lesser than 26.
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Very, very temptingly correct answer!!! But sadly, I found a catch as follows:

Statement 2 says the diagonal of the pentagon is LESS THAN 8 cm.
At the upper limit of this length, the diameter of the circle must therefore be AT LEAST 8 cm, i.e., MORE THAN 8 cm.

The catch is that it may in fact be MUCH MORE than 8 cm, say 10 cm.
In this case, the circumference of the circle will be more than 31 cm.
Therefore, it is possible for the perimeter of the pentagon to be more than 25 cm.

Without knowing HOW MUCH MORE the diameter can be, we can't say for sure.
So we are back to the situation in which we need to know the actual ratio of the length of the diagonal of an inscribed pentagon to the diameter of the circle.
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A pentagon with 5 sides of equal length and 5 interior angles of equal 18 Jun 2023, 00:32
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itspC
1. R= 4 . Circumference of circle is 25.12. so obviously Pentagon perimeter will be less than that as it's inscribed in circle.

2. Diagonal length less than 8 cms . What is intended here is : let's take the largest diagonal whose length is approx. 8 cm we can some how still find the lengths of sides using sine rule --> a/sinA = b/sinB=c/sinC.. as it forms a iscosceles triangle with 108-36-36 --> a=b=a and c=7.999~=8 (say).

So as it's a data sufficiency question without solving overly we can quickly tick off this one too..


So Choose D .


But need to get this thought under 2 minutes is quite challenging task but somehow I would mark D if I am on this question on exam day without wasting time as I know I can't think so much in 2 mins. I would be thinking why specifically 8 and imagine that (here all diagonals of Pentagon are chords of circle) chord length will always be less than diameter length and assume the diameter to be 8 and radius to be 4 i.e., the same as 1st option.


I DON'T MIND TO GET A KUDOS.

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I would go for this explanation. This is the best and i approached the problem in this way for the statement 2. impossible to solve this question if we go by other methods..
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A pentagon with 5 sides of equal length and 5 interior angles of equal Updated on: 06 May 2024, 12:34
Originally posted by DanTheGMATMan on 06 May 2024, 08:34.
Last edited by DanTheGMATMan on 06 May 2024, 12:34, edited 1 time in total.
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­I think it's simplest to just play off of the circumference of the circle in both cases
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A pentagon with 5 sides of equal length and 5 interior angles of equal 06 May 2024, 10:35
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DanTheGMATMan
­I think it's simplest to just play off of the circumference of the circle in both cases (at least as a benchmark in the second one):
­
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That solution is not correct, in Statement 2, when you conclude that the radius must be even shorter than half of the diagonal. The opposite is true. A diameter is the longest line connecting two points on a circle. The pentagon's diagonal is not the longest line connecting two points on the circle. If the diagonal is exactly 8, the diameter must be greater than 8, and the radius must be greater than 4, not less than 4. When you make your argument about the right triangle, notice that the line running through the middle of that triangle is not a full diameter of the circle, so half of that line (which I agree certainly does need to be less than 4 in length) is not the radius.

There really is not a simple solution to this problem, as I and others have pointed out above, unless you rely on facts GMAT test takers would never need to know (about the golden ratio, say). Fortunately for the Focus test, GMAT test takers don't need to worry about this problem any more!

I'm only posting on GMAT Club these days when people reply to posts I've made.
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A pentagon with 5 sides of equal length and 5 interior angles of equal 27 Apr 2025, 05:35
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Can we use the golden ratio for statement?

For Statement 2, you're given that the length of each diagonal of the pentagon is less than 8 cm. Here's how you can quickly analyze this:
  1. Relationship between the side length and diagonal of a regular pentagon:
    • In a regular pentagon inscribed in a circle, the ratio of the length of a diagonal D to the side length S is constant. Specifically, the diagonal is φ times the side length, where φ (the golden ratio) is approximately 1.618.
    So, D=φ×S
  2. Using the given information (Diagonal < 8 cm):
    • We are told that the length of each diagonal is less than 8 cm,
      assume D=8
      8= φ x S
    • Solving for S:
    S=8/ φ
    =4.94cm
    the side length of the pentagon is approximately 4.94 cm.
  3. Perimeter of the Pentagon:
    • The perimeter of the pentagon is simply 5×S where S is the side length.
    • Since S=4.94, the perimeter of the pentagon is : 5×4.94=24.7cm
    • Therefore, the perimeter can be 24.7 cm at max,that is not greater than 26 cm.

Bunuel
A pentagon with 5 sides of equal length and 5 interior angles of equal measure is inscribed in a circle. Is the perimeter of the pentagon greater than 26 centimeters?

(1) The area of the circle is 16π square centimeters.
(2) The length of each diagonal of the pentagon is less than 8 centimeters.



DS75271.01
OG2020 NEW QUESTION
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A pentagon with 5 sides of equal length and 5 interior angles of equal 27 Apr 2025, 18:14
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MohdSadique
Can we use the golden ratio for statement?
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Yes, absolutely -- I'm sure whoever designed this question had the golden ratio in mind, and using it (or a formula derived from it) is the only fast way to do the problem. The solutions in this thread mostly use a different (longer) approach because the golden ratio is not something test takers need to learn (I've never seen another official question where it would be helpful to know).
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