<p>In the figure above, rectangles PQRS and WXYZ each have a perimeter 12 and are inscribed in the circle. How many other rectangles with perimeter 12 can be inscribed in the circle? </p>
<p>(A): One
(B): Two
(C): Three
(D): Four
(E): More than four</p>
<p>Think of it as rotating the rectangles around the edges of the circle, you can get like an infinite number of them because there’s an infinite number of locations on the circle on which your points can be located.</p>
<p>ummm… i think you can just infinitely rotate those figures around the circle. kinda like a diameter(but not literally)? seems like they would all fit.</p>
<p>Yeah i thought about that explanation but it seem so unsubstantiated and weak. I mean what does the perimeter 12 have to do with anything? You can say it can rotate infinately but i can equally say that it the rectangle wont fit at a certain angle. Im trying to find a more substantaiated and clear explanation to this. It just doesnt feel right to have a “it looks like…” or “i think…” answer for a SAT math question</p>
<p>“i can equally say that it the rectangle wont fit at a certain angle”</p>
<p>Of course, you could say that… but you’d be wrong. The fact that the perimeter is twelve has really nothing to do with the answer. You just have to be able to visualize the figure in your head and imagine the rectangles at other angles inside the circle.</p>
<p>I honestly can’t give a more concrete answer than that…</p>
<p>In your mind, rotate the circle 45 degrees (so that the two inscribed rectangles form an ‘X’). You can now inscribe two more rectangles of the same size such that one “goes” north/south and the other west/east. Now rotate the circle again. You can do this an infinite number of times.</p>
<p>The rotating is just to make it easier for you to visualize being able to fit more rectangles in the circle.</p>
<p>when you rotate the figure (with the rectangles inside) the circle is congruent but the rectangles are in a different position…unless by “other rectangles” the person who wrote this question meant different, incongruent ones</p>
<p>I don’t understand why you think rotating the rectangle seems inconcrete. It certainly can be done and if it’s hard to visualize, try drawing a circle and rotating a rectangle in it.
Other than rotating the rectangle you can also change the dimensions of the rectangle and still get infinite number of rectangles. You can have a square with sides of 3 in the circle, for example.
However, the author’s indicating that a simple rotation of the rectangle PQRS can yield a rectangle WXYZ (or vice versa), clearly signals us that to rotate the already given rectangle(s) is the most direct and easy solution to the problem.</p>
<p>Well if you don’t like the idea of rotating the rectangle, try rotating the circle. Since the circle would look the same no matter how much you rotate it (by rotational symmetry), you can say that an infinite number of rectangles would work.</p>
<p>@Bilguun
I don’t think that a square of side length 3 would work. I mean think about it, say that your rectangle with perimeter 12 is very very thin, like infinitely thin. Then it’s length would be 6 and the diameter of the circle would be 6. Then the side length of the square would be 6/sqrt(2), not 3. Of course, it doesn’t really have to do anything with the problem but it shows that you can’t just assume things like that.</p>
<p>This is the dumbest question ever. I remember seeing it. When I saw the question originally, I thought that the answer was too obvious for a high level question, because obviously you can put infinite rectangles in the circle. Yeah, dumb question; easy concept.</p>
