A rectangle is inscribed in a circle of radius r. If the rectangle is not a square, which of the following could be equal to the perimeter of the rectangle?
A. 2r(Sqrt 3)
B. 2r(Sqrt 3 + 1)
C. 4r(Sqrt 2)
D. 4r(Sqrt 3)
E. 4r(Sqrt3 + 1)
[spoiler]OA: B. [/spoiler]
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Rectangle in a circle - tough one
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karthikchandru
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Good question.
If the rectangle is not a square, it means that the if we equally divide the square into 2 rectangles, then their angles will not be 45 degrees (or they won't be 45-45-90 triangles).
However, they can be 30-60-90 triangles. The diagonal of this rectangle will pass through the origin of the circle and hence will have a measure of "2r". We know that in a 30-60-90 triangle, the ratio of the sides is 1:sqrt(3):2
So, if the hypotenuse is "2r", the sides will have be "1r" and sqrt(3)*r".
Therefore, the perimeter of this rectangle will be 2 * (r + sqrt(3)*r) = 2r(Sqrt(3) + 1) --> which is (B)
If the rectangle is not a square, it means that the if we equally divide the square into 2 rectangles, then their angles will not be 45 degrees (or they won't be 45-45-90 triangles).
However, they can be 30-60-90 triangles. The diagonal of this rectangle will pass through the origin of the circle and hence will have a measure of "2r". We know that in a 30-60-90 triangle, the ratio of the sides is 1:sqrt(3):2
So, if the hypotenuse is "2r", the sides will have be "1r" and sqrt(3)*r".
Therefore, the perimeter of this rectangle will be 2 * (r + sqrt(3)*r) = 2r(Sqrt(3) + 1) --> which is (B)
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GMATGuruNY
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When shapes overlap, look for what they have IN COMMON.rahulvsd wrote:A rectangle is inscribed in a circle of radius r. If the rectangle is not a square, which of the following could be equal to the perimeter of the rectangle?
A. 2r(Sqrt 3)
B. 2r(Sqrt 3 + 1)
C. 4r(Sqrt 2)
D. 4r(Sqrt 3)
E. 4r(Sqrt3 + 1)
[spoiler]OA: B. [/spoiler]
When a rectangle is inscribed in a circle, the DIAGONAL of the rectangle is also the DIAMETER of the circle.
Almost every answer choice here includes √3.
√3 implies a 30-60-90 triangle.
The sides of a 30-60-90 triangle are in the following ratio:
1 - √3 - 2.
Let r=1, implying a diameter of 2.
Draw the following figure:
The perimeter of the rectangle above = 2 + 2√3. This is our target.
Now we plug r=1 into the answers to see whether one of them yields our target of 2 + 2√3.
Answer choice B:
2r(√3 + 1) = 2(1)(√3 + 1) = 2√3 + 2.
Success!
The correct answer is B.
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Joseph_Alexander
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I think what distracted me while solving this question was the statement, "if the rectangle is not square". I knew that the diagonal of a square will certainly be the diameter of the circle. But, now when I drawing inscribed rectangles in a circle, I notice that the diagonal of rectangle will certainly be the diameter of a circle.GMATGuruNY wrote:When a rectangle is inscribed in a circle, the DIAGONAL of the rectangle is also the DIAMETER of the circle.
GMAC often says that in a PS question, every bit of data is useful. Why is the data "if the rectangle is not square" given in this question?
Is it just a helper to say the triangles formed won't be a 45-45-90 triangle and to use the answer choices and go ahead presuming the triangle to be a 30-60-90 triangle? :roll:
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Mathsbuddy
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Each diagonal of the rectangle is equal to the diameter of the circle. In addition Circle Theory says that a triangle in a semi-circle is right angled. Therefore the diameter is also the hypotenuse.
The hypotenuse is 2R and we need to find an answer that corresponds with a^2 + b^2 = (2R)^2 [by Pythagoras' Theorem], where and b are the lengths of the other 2 sides.
Then the perimeter will be P = a + b + 2R = 2R([a+b]/2R + 1)
Only one answer corresponds to such a pattern. That is ANSWER B = 2R(Sqrt 3 + 1)
The hypotenuse is 2R and we need to find an answer that corresponds with a^2 + b^2 = (2R)^2 [by Pythagoras' Theorem], where and b are the lengths of the other 2 sides.
Then the perimeter will be P = a + b + 2R = 2R([a+b]/2R + 1)
Only one answer corresponds to such a pattern. That is ANSWER B = 2R(Sqrt 3 + 1)
