In the above diagram, a square is inscribed in a circle, which is insc

Dear Bunuel,

I suppose it is possible to answer the question even without either of the given two statements.

The yellow area = (pi -2) * (r^2)/4 and green area = (r^2)/4

In such a case, yellow is greater than green area for any value of r because (pi-2) > 1


My question, therefore, is: What purpose do the two given statements serve in this question?

correct ans D
1. from statement one --- area of outer square is given .
Side of square can be calculated ---- > similarly radius of circle can be calculated ----> similarly side of inner square .

Now area of yellow = 1/4( area of circle - area of inner square)
area of green = 1/8(area of inner square)

we will get a definite Yes or NO . So 1 st. is ok .

2 . area of green part is given
green part =1/8(Area of inner square)
hence we will follow the reverse process followed in statement one.
Side of square can be calculated <---- similarly radius of circle can be calculated <-----similarly side of inner square .
Hence we find the area of yellow part .

we will get a definite Yes or NO .So 2 st. is ok .
Hence D
_________________

IMO:D

St 1: The area of the larger square is 64.

Side of larger Square = 8 = Diameter of the circle = Diagonal of the smaller square
Radius of circle = 4
Side of smaller square = 4\(\sqrt{2}\)

Area of Yellow region = \(\frac{(Area of circle - Area of Smaller Square)}{4}\)

Are of Green Region = 1/8 * Area of smaller Square

Thus we have all the values hence we can find out which area is greater.
Hence Suff

St 2: The area of the green region is 4.

From this we can find out the Side of the Smaller Square --> Radius of the circle --> Side of the larger Square.
Since we have all the values. It is suff to find out the values and evaluate.
Hence Suff

GROCKIT OFFICIAL SOLUTION:

Let’s start with Statement 1.

A(large square) = 64
s(large square) = 8
diameter = 8
r = 4

First, recognize that the figure is symmetrical. So while we may not explicitly be given an angle to find the sector area (not drawn) in which the yellow region resides, we do know its measure. The diagonals of a square intersect at a right angle, so we can deduce that the sector including the yellow region is 1/4 of the area of the circle. Since we know the radius…

A(large sector) = 1/4 * 16π = 4π

To find the yellow region itself, we must subtract the imaginary (not drawn) triangle from 4π. (Note that this imaginary triangle will be twice the green triangle.)

A(imaginary triangle) = 1/2 * r * r = 1/2 * 4 * 4 = 8

A(Yellow Region) = 4π – 8

The area of the green region can be found in two ways. Either we can see that it’s simply one-half of the 8 we just found, OR we can find both sides of the green triangle with the common 45-45-90 1:1:√2 formula. With a hypotenuse of 4, we derive 2√2 for each side, which yields an area of 4.

Which is greater, 4π – 8 or 4?

4π – 8. Sufficient. (We can save the calculations on the GMAT for DS questions, but it’s still good to go through it to practice for PS questions involving similar calculations.)

What about Statement 2?

If we know the area of the green triangle equals 4, and that it is an isosceles right triangle, then we can set up a simple equation to find its sides, which can be denoted x:

1/2 * x² = 4
x² = 8
x = 2√2

If x = 2√2, then the hypotenuse (r) = 2√2 * √2 = 4. From here, we follow the same logic as we did for Statement 1, and determine Statement 2 is sufficient.

The answer choice is D.
_________________

How do we interpret that "d" is center and "r" is radius????

This is a PS question. I didn't even use the two statements. Could solve it directly by taking variables, as length of sides and radius.

Prompt analysis
A square is inscribed in a circle, which is inscribed in another square. The triangle has one vertice on the diameter of the circle( not necessarily be the centre of the circle), one on the side of the inner square and one as a vertice of the inner square.

Superset
The answer would be green >,=,< yellow

Translation
In order to find the answer, we need:
1#Exact dimensions of the lines and circle
2# Relation between the area of green and yellow
3# location of the vertice of the green triangle on the diameter d.

Statement analysis

St 1: area of larger square is 64. Therefore the side of larger square is 8. Therefore the diameter is 8. Area of segment can be calculated which is yellow in colour by subtracting area of triangle from the area of sector. But we cannot find the area of green triangle as we don't know in which ratio the point, which is the vertice is dividing the diameter d. Hence the area of green cannot be calculated. INSUFFICIENT

St 2: Area of green region is 4. But tha cannot give us the exact dimension of the triangle and hence cannot be correlated to the rest of the figure. INSUFFICIENT

St 1 & St 2: from st1 we know the exact value of the area of the yellow region and for st 2 we know the exact value of the green region. Hence we can compare and get the answer.

Option C

Prompt analysis
A square is inscribed in a circle, which is inscribed in another square. The triangle has one vertice on the diameter of the circle( not necessarily be the centre of the circle), one on the side of the inner square and one as a vertice of the inner square.

Superset
The answer would be green >,=,< yellow

Translation
In order to find the answer, we need:
1#Exact dimensions of the lines and circle
2# Relation between the area of green and yellow
3# location of the vertice of the green triangle on the diameter d.

Statement analysis

St 1: area of larger square is 64. Therefore the side of larger square is 8. Therefore the diameter is 8. Area of segment can be calculated which is yellow in colour by subtracting area of triangle from the area of sector. But we cannot find the area of green triangle as we don't know in which ratio the point, which is the vertice is dividing the diameter d. Hence the area of green cannot be calculated. INSUFFICIENT

St 2: Area of green region is 4. But tha cannot give us the exact dimension of the triangle and hence cannot be correlated to the rest of the figure. INSUFFICIENT

St 1 & St 2: from st1 we know the exact value of the area of the yellow region and for st 2 we know the exact value of the green region. Hence we can compare and get the answer.

Option C

Hi Bunuel,

Is n't the question itself self-sufficient?, as we know the order how the figures are inscribed, we will be able to say which is greater yellow or green without the help of the statements.

Please correct me if am missing anything.

Hi Bunuel. I have the same doubt , can you please clarify.

The question indeed appears to be self sufficient. If it were asked, for example "What is the difference between the area of the green part and the yellow part?" then it would be more appropriate.

Posted from my mobile device

Exactly, for the solution I did not need any of these two info.

How do we say that d is the centre