A Pentagon With 5 Sides Of Equal Length And 5 Interior Angles Of Equal measure GMAT Data Sufficiency

Question: 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 centimetres?

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

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are not sufficient.

Correct Answer: D

Solution and Explanation:
Approach Solution 1:
The problem statment states that:
Given:

A pentagon with 5 sides of equal length and 5 interior angles of equal measure is inscribed in a circle.

Asked:

Find out if the perimeter of the pentagon is greater than 26 centimetres.

Considering the 1st statement, it highlights that the area of the circle is 16\(\Pi\) square centimetres.
It can be concluded from the first statement that the radius of the circle is 4. Thus, this establishes that the size of the inscribed pentagon is locked into an exact single shape. Further, this highlights that the perimeter of the inscribed pentagon can hold a single value.
Thus,with the help of some basic school trigonometry, we can find the perimeter. However, we COULD even just draw a circle with radius 4. Further, 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 can conclude answer the target question with certainty, statement 1 is sufficient.

Nevertheless, considering the second statement, it says that the length of each diagonal of the pentagon is less than 8 centimetres.
The rule that is applicable in this case is the sum of 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:

Let’s consider focusing upon the red and blue triangles as shown below:

Considering both the triangles, they portray the same angles and share the same diagonal, both triangles are congruent triangles.
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 centimetre
Since we can answer the target question with certainty, statement 2 is sufficient.

Therefore, EACH statement ALONE is sufficient.

Approach Solution 2:
The problem statment states that:
Given:

A pentagon with 5 sides of equal length and 5 interior angles of equal measure is inscribed in a circle.

Asked:

Find out if the perimeter of the pentagon is greater than 26 centimetres.

Statement 1:The area of the circle is 16π square centimetres.

Therefore, the statement implies exactly how big the circle is. Hence we know exactly how big the pentagon is. Thus we can compute its exact perimeter and answer the question.
Hence, statement 1 alone is sufficient.

Statement 2: The length of each diagonal of the pentagon is less than 8 centimetres.

If you draw the three diagonals that is drawn in the picture below, and label all of the relevant angles, we will see that the triangles labeled '1', '2' and '3' all are 36-36-108 triangles.

Triangles 1 and 2 are identical ('congruent') because they have the same angles, and they share a side (opposite the 108 degree angle in both). So the sides 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. This is because each is just part of a sloping diagonal of the pentagon but not counting the 's' part that belongs to triangle 2. Therefore 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.
Therefore 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.
Hence, Statement 2 is also sufficient.

Diagram:

Therefore, EACH statement ALONE is sufficient.

Approach Solution 3:
The problem statment states that:
Given:

A pentagon with 5 sides of equal length and 5 interior angles of equal measure is inscribed in a circle.

Asked:

Find out if the perimeter of the pentagon is greater than 26 centimetres.

Statement (1) : The area of the circle is 16π square centimetres.
This statement pins down the radius of the circle. Hence there lies only one way to draw the pentagon. The perimeter will be definitively greater than 26cm or not. Thankfully, it is not required to determine whether it is or isn’t. Hence statement (1) is sufficient.

Statement (2) The length of each diagonal of the pentagon is less than 8 centimetres.
This statement demands to 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. We need to figure out whether the perimeter is > 26cm and 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. 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

d/s=s/(d–s)
This yields
s²=d(d–s)=d²–ds
Since we’re trying to figure out what d < 8 tells us about s, we can set that up as:
s²= d²– ds<8²– 8s
So the following must be true
s²<8²–8s
Re-arranging, the following must be true
s²+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, s²+8s strictly increases as s increases. Therefore, any s > 5 will also yield a contradiction. Therefore s is definitively NOT greater than 5.2. Hence statement (2) is sufficient.

Diagram:

Therefore, EACH statement ALONE is sufficient.

“A pentagon with 5 sides of equal length and 5 interior angles of equal measure”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This question has been taken from the book “GMAT Official Guide 2022”. The GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions come up with a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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A Pentagon With 5 Sides Of Equal Length And 5 Interior Angles Of Equal measure GMAT Data Sufficiency

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