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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
(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
Pure algebraic questions like this one are no longer part of the GMAT Data Sufficiency syllabus.
For relevant Data Sufficiency questions, please check the Data Sufficiency forum:
Data Sufficiency Forum
Data Sufficiency Forum
Updated on: 06 May 2019, 00:05
Originally posted by DavidTutorexamPAL on 26 Apr 2019, 14:42.
Last edited by DavidTutorexamPAL on 06 May 2019, 00:05, edited 3 times in total.
Last edited by DavidTutorexamPAL on 06 May 2019, 00:05, edited 3 times in total.
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*** The following solution has been revised thanks to Ian's comment below. Thanks, Ian.
The Alternative approach to this question - since we know very little about a regular pentagon - is to try using specific numbers and more familiar shapes to assist us.
Statement (1): we can use the formula of circular area and find that the radius is 4. Now, if it were a regular HEXAGON inscribed in the circle, its side would have been equal to the radius (since it can be divided into 6 equilateral triangles), and thus its perimeter would have been 6x4=24. But since the more sides there are to a regular polygon inscribed in a circle, the larger its perimeter, then the perimeter of the pentagon must be less than 24. That's enough. Answer choices (B), (C) and (E) are eliminated.
Statement (2): As the attached figure shows, the green and the red triangles created by the diagonals are identical (that's because the pentagon's 108 angle is divided into three equal angles - opposite the same arc - of 36 degrees, so all of their angles are equal; and their largest angle is opposite the same side - the diagonal). Thus, since if the pentagon's perimeter was 26 each side would have been just a bit above 5, we'll mark all sides as 5 to see whether this is possible. Also, since the statement relates to the diagonal as less than 8, we'll try 8. If that's true, the sides of the blue triangle are 8 - 5 = 3. Now, how do we know if that's possible? The green and the blue triangles are similar triangles (having 2 identical angles of 36 degrees), so
3/5 = 5/8 We'll multiply by 5 & 8:
24 = 25
This is wrong, of course, and since the diagonal is smaller than 8 and the side is larger than 5, the left side of the equation is even smaller than 24. Thus, in order to create a correct equation, the side of the pentagon must be smaller than 5, and so its perimeter is smaller than 5x5=25 >>> smaller than 26.
So statement (2) also gives us enough information, and the correct answer is (D).
The Alternative approach to this question - since we know very little about a regular pentagon - is to try using specific numbers and more familiar shapes to assist us.
Statement (1): we can use the formula of circular area and find that the radius is 4. Now, if it were a regular HEXAGON inscribed in the circle, its side would have been equal to the radius (since it can be divided into 6 equilateral triangles), and thus its perimeter would have been 6x4=24. But since the more sides there are to a regular polygon inscribed in a circle, the larger its perimeter, then the perimeter of the pentagon must be less than 24. That's enough. Answer choices (B), (C) and (E) are eliminated.
Statement (2): As the attached figure shows, the green and the red triangles created by the diagonals are identical (that's because the pentagon's 108 angle is divided into three equal angles - opposite the same arc - of 36 degrees, so all of their angles are equal; and their largest angle is opposite the same side - the diagonal). Thus, since if the pentagon's perimeter was 26 each side would have been just a bit above 5, we'll mark all sides as 5 to see whether this is possible. Also, since the statement relates to the diagonal as less than 8, we'll try 8. If that's true, the sides of the blue triangle are 8 - 5 = 3. Now, how do we know if that's possible? The green and the blue triangles are similar triangles (having 2 identical angles of 36 degrees), so
3/5 = 5/8 We'll multiply by 5 & 8:
24 = 25
This is wrong, of course, and since the diagonal is smaller than 8 and the side is larger than 5, the left side of the equation is even smaller than 24. Thus, in order to create a correct equation, the side of the pentagon must be smaller than 5, and so its perimeter is smaller than 5x5=25 >>> smaller than 26.
So statement (2) also gives us enough information, and the correct answer is (D).
Attachments
GC_circumscribed_2.png [ 46.52 KiB | Viewed 111276 times ]
Updated on: 07 May 2022, 06:01
Originally posted by IanStewart on 28 Apr 2019, 12:07.
Last edited by IanStewart on 07 May 2022, 06:01, edited 1 time in total.
Last edited by IanStewart on 07 May 2022, 06:01, edited 1 time in total.
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Statement 1 is immediately sufficient, because if we know exactly how big the circle is, we know exactly how big the pentagon is, so we can compute its exact perimeter and answer the question.
If there's an extremely fast way to prove Statement 2 is sufficient as written, I don't see it (the solution someone else posted above isn't correct edit: because gmatclub reorders posts, the solution I was referring to when I originally wrote this is no longer the one "above" mine, it's somewhere below mine now), though it would be much easier with slightly different numbers. If you draw the three diagonals I draw in the picture below, and label all of the relevant angles, you'll see that the triangles I've labeled '1', '2' and '3' all are 36-36-108 triangles. Triangles 1 and 2 are identical ('congruent' in math-speak) because they have the same angles, and they share a side (opposite the 108 degree angle in both). So the sides I've labeled with blue tickmarks are all equal. Triangle 3 is similar to, but smaller than, triangles 1 and 2.
If we call the side of the pentagon 's' and the diagonal 'd', then the sides of triangle 1 are just s, s, and d. Notice that in triangle 3, the two equal sides, the sloping ones, are both d-s in length (because each is just part of a sloping diagonal of the pentagon, but not counting the 's' part that belongs to triangle 2, so we just need to subtract that). So the sides of triangle 3 are d-s, d-s, and s. Since triangle 1 and triangle 3 are similar, their sides are in the same ratio. So if we divide the longer side by the shorter side in triangle 1, we get the same ratio as we get when we do the same for triangle 3:
d/s = s/(d-s)
And rewriting this:
d^2 - sd = s^2
d^2 = s^2 + sd
Now say d = 8, the largest the diagonal is allowed to be according to Statement 2. Then plugging in,
64 = s^2 + 8s
and now you can see if you just plug in s = 5, the right side of this equation is very slightly too big. So s cannot be 5 -- the equation will only work if s is slightly less than 5. And if the diagonal is shorter than 8, then s will need to be even smaller of course. In any case, the perimeter is certainly less than 25. So Statement 2 is also sufficient, and the answer is D.
If there's an extremely fast way to prove Statement 2 is sufficient as written, I don't see it (the solution someone else posted above isn't correct edit: because gmatclub reorders posts, the solution I was referring to when I originally wrote this is no longer the one "above" mine, it's somewhere below mine now), though it would be much easier with slightly different numbers. If you draw the three diagonals I draw in the picture below, and label all of the relevant angles, you'll see that the triangles I've labeled '1', '2' and '3' all are 36-36-108 triangles. Triangles 1 and 2 are identical ('congruent' in math-speak) because they have the same angles, and they share a side (opposite the 108 degree angle in both). So the sides I've labeled with blue tickmarks are all equal. Triangle 3 is similar to, but smaller than, triangles 1 and 2.
If we call the side of the pentagon 's' and the diagonal 'd', then the sides of triangle 1 are just s, s, and d. Notice that in triangle 3, the two equal sides, the sloping ones, are both d-s in length (because each is just part of a sloping diagonal of the pentagon, but not counting the 's' part that belongs to triangle 2, so we just need to subtract that). So the sides of triangle 3 are d-s, d-s, and s. Since triangle 1 and triangle 3 are similar, their sides are in the same ratio. So if we divide the longer side by the shorter side in triangle 1, we get the same ratio as we get when we do the same for triangle 3:
d/s = s/(d-s)
And rewriting this:
d^2 - sd = s^2
d^2 = s^2 + sd
Now say d = 8, the largest the diagonal is allowed to be according to Statement 2. Then plugging in,
64 = s^2 + 8s
and now you can see if you just plug in s = 5, the right side of this equation is very slightly too big. So s cannot be 5 -- the equation will only work if s is slightly less than 5. And if the diagonal is shorter than 8, then s will need to be even smaller of course. In any case, the perimeter is certainly less than 25. So Statement 2 is also sufficient, and the answer is D.
Attachments
Screen Shot 2019-04-28 at 3.02.23 PM.png [ 31.01 KiB | Viewed 116321 times ]
