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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
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01 Jun 2020, 07:53
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Bunuel
Here's a solution that, in my opinion, combines all of the best (i.e., fastest) approaches demonstrated by other posters in this thread.
Given: A pentagon with 5 sides of equal length and 5 interior angles of equal measure is inscribed in a circle.
Target question: Is the perimeter of the pentagon greater than 26 centimeters?
Statement 1: The area of the circle is 16π square centimeters.
IMPORTANT: For geometry Data Sufficiency questions, we're typically checking to see whether the statements "lock" a particular angle, length, or shape into having just one possible measurement. This concept is discussed in much greater detail in the following video: https://www.gmatprepnow.com/module/gmat-geometry/video/884
From statement 1, we can conclude that the radius of the circle is 4. This means the size of the circle and the size of the inscribed pentagon are LOCKED into to exactly one shape, which means the perimeter of the inscribed pentagon can have only one value.
So, we COULD apply some high school trigonometry to find the perimeter, or we COULD even just draw a circle with radius 4, then draw an inscribed pentagon, and then physically measure the perimeter. Regardless of what technique we use, we can definitely determine whether the perimeter of the pentagon is greater than 26 cm
Since we COULD answer the target question with certainty, statement 1 is SUFFICIENT
Statement 2: The length of each diagonal of the pentagon is less than 8 centimeters.
That statement is much trickier!
Useful rule: the sum of the angles in an n-sided polygon = (n - 2)(180°)
So the sum of the angles in the pentagon = (5 - 2)(180°) = 540°
Since each of the 5 angles are equivalent, the measurement of each angle = 540°/5 = 108°
There are 2 diagonals at each vertex. Each 2 diagonals divide the 108° into 3 equivalent angles of 36°
So we can derive the following angles:
Now focus on the red and blue triangles below.
Since both triangles have the same angles AND share the same diagonal, both triangles are congruent (aka identical)
So if we let x = the length of each side of the pentagon, we know that the two sides of the blue triangle must also have sides of length x
We are told that the length of each diagonal is less than 8.
So let's see what happens when the length of each diagonal is exactly 8.
This means the length of AD = 8
So, if AP = x, then PD = 8-x
We can apply the same logic to show that PE = 8-x
At this point we need only recognize that ∆ABC is similar to ∆EPD
Since the two triangles are similar, the ratios of their corresponding sides must be equal
This means: 8/x = x/(8-x)
Cross multiply to get: (x)(x) = (8)(8 - x)
Simplify: x² = 64 - 8x
Add 8x to both sides to get: x² + 8x = 64
ASIDE: At this point we COULD set the above equation equal to zero, and then try to solve the quadratic equation. Unfortunately the resulting quadratic equation is not easily factored, which means we have to apply the quadratic formula. However, instead of applying the quadratic formula, let's test a possible value of x.
Let's test x = 5.
Plug this value into our equation to get: 5² + 8(5) = 64
Evaluate: 65 = 64
As we can see, x = 5 is NOT a solution to the equation x² + 8x = 64
More importantly, we can see that, in order to satisfy the equation, x must be less than 5
If x is less than 5, then the perimeter of the pentagon must be less than 25
So, the answer to the target question is NO, the perimeter of the pentagon is NOT greater than 26 centimeter
Since we can answer the target question with certainty, statement 2 is SUFFICIENT
Answer: D
Cheers,
Brent
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.
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Screen Shot 2019-04-28 at 3.02.23 PM.png [ 31.01 KiB | Viewed 116306 times ]
25 May 2019, 11:33
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I agree with how Ian is looking at this – definitely prefer this approach over analyzing arc lengths for statement (1) and solving the quadratic equation \(d^{2} – ds – s^{2} = 0\) with the quadratic formula for statement (2), as the official solution suggests.
Here are a few more thoughts on how to get through this beast!
Statement (1) pins down the radius of the circle, so there’s only one way to draw the pentagon, and the perimeter will be definitively greater than 26cm or not. Thankfully, we don’t have to determine whether it is or isn’t – statement (1) is sufficient.
Statement (2) demands that we determine the relationship between the side s and diagonal d in a regular pentagon. This will help us figure out what d < 8cm tells us about s. That’s driving the analysis. Of course, to figure out whether the perimeter is > 26cm, we need to figure out whether s > 5.2.
First, this definitely requires some good sketching! In the figure below, we label what we can to piece together a relationship between d and s. Here’s how you can determine the angles in the diagram.
1. There are (5 – 2) x 180˚ = 540˚ in any pentagon. In this case, since all interior angles are equal, each interior angle = 108˚.
2. Any central angle formed by connecting adjacent vertices of the pentagon to the center of the circle must be 1/5 of 360˚ = 72˚.
3. Each of the angles labeled 36˚ is an inscribed angle, and therefore equal to half of the corresponding 72˚ central angle.
4. Of course, we also need to use the fact that triangles have a total of 180˚, and opposite angles are equal.
Next, because the angles in triangle 1 match the angles in triangle 2, these two triangles are similar, and because the ratio of their sides is 1:1, these two triangles are congruent. Because the angles in triangle 3 match the angles in triangles 1 and 2, triangle 3 is similar to triangles 1 and 2. As a result, corresponding sides are in a constant ratio, and we get
\(\frac{d}{s} = \frac{s}{d – s}\)
This yields
\(s^{2} = d(d – s) = d^{2} – ds\)
Since we’re trying to figure out what d < 8 tells us about s, we can set that up as:
\(s^{2} = d^{2} – ds < 8^{2} – 8s\)
So the following must be true
\(s^{2} < 8^{2} – 8s\)
Re-arranging, the following must be true
\(s^{2} + 8s < 64\)
Since we’re trying to see whether s > 5.2cm, we can test a case near the borderline (s = 5cm) to get some insight.
If we assume s = 5, we get 25 + 40 < 64, which is a contradiction. Therefore, the assumption that s = 5 must be false.
Since s must be positive, \(s^{2} + 8s\) strictly increases as s increases. Therefore, any s > 5 will also yield a contradiction. Therefore s is definitively NOT greater than 5.2, and statement (2) is sufficient.
Answer: D
Here are a few more thoughts on how to get through this beast!
Statement (1) pins down the radius of the circle, so there’s only one way to draw the pentagon, and the perimeter will be definitively greater than 26cm or not. Thankfully, we don’t have to determine whether it is or isn’t – statement (1) is sufficient.
Statement (2) demands that we determine the relationship between the side s and diagonal d in a regular pentagon. This will help us figure out what d < 8cm tells us about s. That’s driving the analysis. Of course, to figure out whether the perimeter is > 26cm, we need to figure out whether s > 5.2.
First, this definitely requires some good sketching! In the figure below, we label what we can to piece together a relationship between d and s. Here’s how you can determine the angles in the diagram.
1. There are (5 – 2) x 180˚ = 540˚ in any pentagon. In this case, since all interior angles are equal, each interior angle = 108˚.
2. Any central angle formed by connecting adjacent vertices of the pentagon to the center of the circle must be 1/5 of 360˚ = 72˚.
3. Each of the angles labeled 36˚ is an inscribed angle, and therefore equal to half of the corresponding 72˚ central angle.
4. Of course, we also need to use the fact that triangles have a total of 180˚, and opposite angles are equal.
Next, because the angles in triangle 1 match the angles in triangle 2, these two triangles are similar, and because the ratio of their sides is 1:1, these two triangles are congruent. Because the angles in triangle 3 match the angles in triangles 1 and 2, triangle 3 is similar to triangles 1 and 2. As a result, corresponding sides are in a constant ratio, and we get
\(\frac{d}{s} = \frac{s}{d – s}\)
This yields
\(s^{2} = d(d – s) = d^{2} – ds\)
Since we’re trying to figure out what d < 8 tells us about s, we can set that up as:
\(s^{2} = d^{2} – ds < 8^{2} – 8s\)
So the following must be true
\(s^{2} < 8^{2} – 8s\)
Re-arranging, the following must be true
\(s^{2} + 8s < 64\)
Since we’re trying to see whether s > 5.2cm, we can test a case near the borderline (s = 5cm) to get some insight.
If we assume s = 5, we get 25 + 40 < 64, which is a contradiction. Therefore, the assumption that s = 5 must be false.
Since s must be positive, \(s^{2} + 8s\) strictly increases as s increases. Therefore, any s > 5 will also yield a contradiction. Therefore s is definitively NOT greater than 5.2, and statement (2) is sufficient.
Answer: D
Attachments
OG2020 DS369.jpg [ 44.3 KiB | Viewed 111749 times ]
