Inscribed Square - Geometry Question Bank

[debmalya.dutta]

This question is from the from Manhattan GMAT Geometry question bank

I am confused here . The correct answer per the answer key is C.

Ron/Stacey / Manhattan Gmat Tutors,
Question :
can you please provide a clarification
Rectangle ABCD is inscribed in circle P. What is the area of circle P?
(1) The area of rectangle ABCD is 100.
(2) Rectangle ABCD is a square.

Statement 1
suppose the sides of rectangle are x & y
so ...xy = 100

now in triangle adc ,
ad = x = 2r sin 45 where r is the radius of the circle
cd = y= 2r cos 45 where r is the radius of the circle

xy = 4 r^2 sine 45 cos 45

since we know xy = 100 and sin 45=cos 45 = 1/ root of 2
we can determine the value of r

and hence find the area of the circle.

Please let me know where I went wrong on this one.. Must be missing some fine point

[loving.achin]

Hello debmaiya.dutta,
The logic you applied is perfectly valid (__only__ if you take that rectangle to be a square. i.e. the angle you took 45 degree in 2rsin45 and 2rcos45 is only valid if that rectangle is a square).

To illustrate the above, draw two rectangles with the same area 100. For eg: draw a rectangle with sides (50 and 2) and (20 and 5) and see if you can prove 45 degree concept. Surely you won't.

Hence you need statement (2) as well and hence the solution should be 'C'

Thanks
Achin

[debmalya.dutta]

Right.. thanks .. forgot that aspect... thanks again

[mschwrtz]

ty loving.achin

another excellent post

[mschwrtz]

Oh, by the way, you'll never need trig on the GMAT. You have to work pretty hard to find a question where it's even helpful.

And, of course, if the rectangle were a square, you could solve this more easily without trig.

[nicole.laurene.dong]

Hi there - if we were to create any rectangle that has all 4 vertices touching the circle of P and has an area of 100, won't the vertical line (hypotenuse) always be the diameter of the circle, no matter what the size the height and width of the rectangle? I'm a bit confused.

I understand that different areas for rectangles develop different lengths of lines, however, given that the rectangles vertices all touch the circle - and the circle is the same size, wouldn't all hypotenuses be the same?

Thanks in advance.

[RonPurewal]

wouldn't all hypotenuses be the same?

Thanks in advance.

No.

The problem here is that you're imagining one circle, of fixed size, and thinking about drawing various rectangles inside it. That's not what is going on here.

In fact, this is a pretty big issue, because it means that you're not thinking at all about the goal of the problem.
The whole point of the problem is to find whether the circle actually has a fixed size! If you only visualize 1 circle, you're not addressing what's actually at issue in the problem.

Remember, it's the area of the rectangle that is fixed here.

Just try a couple of different rectangles with area = 100, and it will soon become very clear that you can have all different sizes of circles.
If you think about extreme examples, you won't even have to calculate anything to see this. E.g., consider a 10 x 10 square, and then a 1 x 100 "stick" rectangle; it should be obvious that the circle is much, much larger in the second case.

[shikhers747]

But ron, where it is written that rectangle touches the circle..
this may be a possibility...
Can you plz explain

[shikhers747]

But ron, where it is written that rectangle touches the circle..
this may be a possibility...
Can you plz explain

Can someone plz answer my query

[RonPurewal]

But ron, where it is written that rectangle touches the circle..
this may be a possibility...
Can you plz explain

this is the definition of "inscribed".

still, this is an uncommon term; the GMAT won't require knowledge of such things.
if this term were to appear on the exam, it would almost certainly be accompanied by a diagram and/or explanation.

[RonPurewal]

Can someone plz answer my query

by making posts like this one, you're delaying the time it takes us to answer your question.

within each folder, we answer questions from oldest to newest. if you "bump" your own thread, you're accomplishing nothing except making it the "newest" (= last) thread again.