A rectangle is inscribed in a circle of diameter 2. Which of

Question

A rectangle is inscribed in a circle of diameter 2. Which of the following can be the area of the rectangle?

I. 0.01
II. 2.00
III. 3.20

A. I only
B. II only
C. III only
D. I and II only
E. II and III only

Bunuel · Accepted Answer

If the length of a diagonal of a square is 2, then side^2+side^2= diagonal^2 --> 2*side^2=2^2 --> side^2=area=2 . You could get the side in another way: since the angle between a diagonal and a side in a square is 45 degrees, then side=\frac{diagonal}{\sqrt{2}} (from the properties of 45-45-90 triangle). You could also get the area directly: area_{square}=\frac{diagonal^2}{2}=2 . Complete solution: A rectangle is inscribed in a circle of diameter 2. Which of the following can be the area of the rectangle? I. 0.01 II. 2.00 III. 3.20 A. I only B. II only C. III only D. I and II only E. II and III only Look at the diagram below: m24-05.png If the width of blue rectangle is small enough then its area could be 0.01. Generally, the are of the inscribed rectangle is more than 0 and less than or equal to the area of the inscribed square (inscribed square has the largest area from all rectangles that can be inscribed in a given circle). Now, since the area of the inscribed square in a circle with the diameter of 2 is 2, then the area of the inscribed rectangle is 0

walker · Answer

Area of a rectangle ranges from 0 to area of square=2. S0, I and II is possible.

GK_Gmat · Answer

Simple enough now! Thanks a bunch Walker.

marcodonzelli · Answer

moreover, the area of the circle is 3,14...thus 3,20 is the area of a figure which is not inscribed....

voodoochild · Answer

Bunuel / Walker,
I have a question. If the diameter = 2 => diagonal of square =2 => 2(side of a square)^2 = 2 => (side of a square)^2 = 1.

Hence, the area of square is 1!

Thoughts?

SOURH7WK · Answer

diagonal of Square = a* root 2=2, Hence a (side of sq)= root 2.

Area of Square = root 2 X root 2 = 2

EvaJager · Answer

Area of a square is also diagonal * diagonal / 2, so in our case is 2 * 2 / 2 =2. If you want to find the side of the square, then don't forget to square: 2(side of a square)^2 = 2 ^2=4 ... Be more careful with your computations. On the real test you won't get help from the forum's members

)

EvaJager · Answer

The link to the figure is not working/missing...

Did you mean something like this? (See the attached figure)

We can make the area of the inscribed rectangle as close as possible to 0, when for example the length is getting closer, and closer to the diameter, while the width is getting closer and closer to 0.

The maximum area can be obtained when the height of the right triangle, which is half of the rectangle and it is inscribed in the half circle, is equal to the radius of the circle. In this case, the rectangle becomes a square.

Bunuel · Answer

It should be visible now.

voodoochild · Answer

Thanks Bunuel...I missed this problem because of a silly mistake.

Thanks again.

WholeLottaLove · Answer

A rectangle is inscribed in a circle of diameter 2. Which of the following can be the area of the rectangle?

If the circle has a diameter of 2, then it's radius is 1 so it's area is pi*1^2 or simply pi. Pi is equal to 3.1415. A a rectangle of .01 or 2.0 could fit inside this circle but not a rectangle with an area larger than the circle.

D

I. 0.01
II. 2.00
III. 3.20

A. I only
B. II only
C. III only
D. I and II only
E. II and III only

AlexeyGevlich · Answer

Rectangle could be a square in this problem? The correct answer is still D, but I'm just curious

Bunuel · Answer

Yes, because all squares are rectangles.

GyMrAT · Answer

Let  x  be the length &  y  be the width of the rectangle. Its diagonal will be the diameter of the circle, as the rectangle is inscribed in the circle.

Therefore length of diagonal =  2  & Area of the rectangle =  xy

Hence we have  x^2 + y^2 = 4

or  (x - y)^2 + 2xy = 4

or  (x - y)^2 = 4 - 2xy

LHS has to be positive or 0, hence  4 - 2xy >=0

We can see that only I & II satisfy the above inequality.

Answer D.

Thanks,
GyM

bumpbot · Answer

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A rectangle is inscribed in a circle of diameter 2. Which of

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A rectangle is inscribed in a circle of diameter 2. Which of

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