The heart-shaped decoration shown in the figure above consists of a sq

Question

https://gmatclub.com/forum/download/file.php?id=47616 The heart-shaped decoration shown in the figure above consists of a square and two semicircles. What is the radius of each semicircle? (1) The diagonal of the square is 10\sqrt{2} centimeters long. (2) The area of the square region minus the sum of the areas of the semicircular regions is 100 − 25π square centimeters. DS18502.01 Quantitative Review 2020 NEW QUESTION 2019-04-26_1840.png

DavidTutorexamPAL · Accepted Answer

Please note that the solution presented above by Chethan92 is wrong, as it relates to statement (2) as an equation with two variables, but relates to it as having only one solution. 
The Logical approach to this question will be based on the fact that in both squares and circles one measurement, any measurement (side, diagonal, radius, perimeter, etc.) is enough on order to find all other measurements.
Statement (1) is enough, since the diagonal of the square can give us its side (since it creates a 45-45-90 triangle, all we need is to divide the diagonal by √2). And since the radius is half the length of this side (which is also the diameter of the semicircle), that's enough information.
Statement (2) provides one equation with one variable, and thus has only one possible solution. We only need one variable since we can use 2r as the side of the square. Also, since in Geometry there are no negative measurements, we can expect just one, positive value of r.
Since each of the statements is sufficient on its own, the correct answer is (D).

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Chethan92 · Answer

From S1:

Diagonal of a square =  a\sqrt{2}  =  10\sqrt{2} 
Side, a = 10
diameter of the semicircle = side of the square. 
Radius = 10/2 = 5
Sufficient.

From S2:

a^2-π*r^2  = 100-25π
 a^2 = 100 
 π*r^2 = 25π 
Radius = 5
Sufficient.

D is the answer.

Chethan92 · Answer

DavidTutorexamPAL , I still don't understand why my solution is wrong. can you elaborate?

DavidTutorexamPAL · Answer

Sure.
This is an equation with two variables, which means that it has an infinite number of possible solutions:
a² − πr² = 100 - 25π
Say, for example, that a = 1:
99 - πr² = 100 - 25π
25π - 1 = πr²
And from here you can get a quite-nasty-non-integer-but-possible value of r.
The reason in this specific question why there is only one possible solution is that:
A. We can substitute a with 2r.
B. In Geometry, lengths must be positive.
So you did get to the correct answer, but for the wrong reasons

I hope this is clear.

Posted from my mobile device

I hope this is clear. Posted from my mobile device

mgpangwi · Answer

Well since 1 side of the square = 2r

For statement 2

(2r)^{2}  -  \pi  r^{2} = 100-25 \pi

r^{2}  (4-  \pi  ) = 25 ( 4-  \pi  )

*cancel out ( 4-  \pi  )

r^{2}  = 25

r = 5

MHIKER · Answer

(1) Diagonal of a square =  s\sqrt{2}  =  10\sqrt{2} ,  s  is a side of the square.

Side,  s = 10

The diameter of the semicircle = side of the square.

Radius  = \frac{10}{2} = 5 ; Sufficient.

(2)   Given,

s^2-π*r^2= 100-25π

s^2-π*r^2= 10^2-5^2π

s^2 = 10^2 , or  π*r^2 = 5^2π

Radius  = 5 ; Sufficient.

The answer is  D

woohoo921 · Answer

KarishmaB If you have time, I would be so appreciative for your thoughts on my question below... I thought that statement (1) was not sufficient because we do not know when we draw the diagonal (turning the square into two triangles), if the triangles are 30-60-90 or 45-45-90 triangles, thereby providing different lengths of the sides. I see that we are able to use the Pythagorean theorem and now understand why it made sense to solve the problem that way. This leads me to ask, would you only use the Pythagorean theorem when you are not sure if it you are dealing with a 30-60-90 or 45-45-90 triangle? Thank you for all of your time and help

Bunuel · Answer

A couple of points to consider:

1. The diagonal of a square always divides it into two  isosceles  (45°-45°-90°) triangles. Two 30°-60°-90° triangles do not form a square because their sides are unequal, and when combined, they produce a rectangle with differing length and width, not a square with equal sides.

2. There's no need for complex calculations here. Since we know the diagonal of the square from point (1), the square is well-defined, allowing us to answer any question about it. We can determine its side length, which will be twice the radius of the semicircle. From (2), we can construct an equation with one unknown both the area of the square and the area of the semicircle can be expressed in terms of the semicircle's radius (the square's side length = 2r). Thus, we can solve the equation: (2r)^2 - πr^2 = 100 − 25π.

zoezhuyan · Answer

dear  Bunuel ,
I can understand statement2 is sufficient , but I have no idea where is the problem that my thought about the insufficiency about statement 2.
when I encountered statement2 of this question, I thought we have no idea what is the area of square, maybe 100, or maybe 100+ 50π, then the area of semicircle is 75π, there are lots of possibilities, so I crossed of B.

appreciate your clarify.

thanks in advance

Bunuel · Answer

We have a shape that consists of a square and two semicircles attached to it. It's crucial to note that the dimensions of the semicircles are connected to the dimensions of the square, and vice versa. This implies that if you scale the entire figure up or down, all parameters will proportionally increase or decrease. Point (2) establishes the relationship between the square's and semicircle's parameters, where the difference in their areas is x. By understanding that only one specific size of the figure results in that area difference of x, we can conclude that the figure's size is fixed. Consequently, knowing the difference in the areas enables us to determine all other parameters of the shape.

Does this make sense?

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The heart-shaped decoration shown in the figure above consists of a sq

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