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# Can a circle be drawn so that its circumference includes all four vert

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Can a circle be drawn so that its circumference includes all four vert  [#permalink]

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21 Dec 2017, 03:33
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75% (hard)

Question Stats:

31% (01:19) correct 69% (00:48) wrong based on 41 sessions

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Can a circle be drawn so that its circumference includes all four vertices of quadrilateral Q ?

(1) All four sides of Q are equal in length.
(2) Each of the interior angles of Q measures 90°.

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Re: Can a circle be drawn so that its circumference includes all four vert  [#permalink]

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21 Dec 2017, 06:39
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Can a circle be drawn so that its circumference includes all four vertices of quadrilateral Q ?

(1) All four sides of Q are equal in length.
(2) Each of the interior angles of Q measures 90°.

Question: Can a circle inscribe a given Quadrilateral?

Statement 1: All four sides of Q are equal in length.

Case 1: In case of square, the answer to question is YES
Case 2: In case of all other Rhombus, the answer to question is NO

NOT SUFFICIENT

Statement 2: Each of the interior angles of Q measures 90°.
Now the quadrilateral is definitely a rectangle (which includes the case of square as well)
Hence answer yo the question is f=definitely YES

SUFFICIENT

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Re: Can a circle be drawn so that its circumference includes all four vert  [#permalink]

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21 Dec 2017, 10:31
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Can a circle be drawn so that its circumference includes all four vertices of quadrilateral Q ?

(1) All four sides of Q are equal in length.
(2) Each of the interior angles of Q measures 90°.

Since we're asked if a circle can be drawn, why not just try? This is a classic Alternative approach.

(1) All four sides of Q are equal in length so it is a square or a more messy rhombus.
We'll try drawing a square inscribed by a circle - yes!
We'll try drawing a rhombus inscribed by a circle - no!
Playing with the figure, we can SEE that this won't work because the diagonals of a rhombus aren't always equal.
Insufficient!
(2) Our quadrilateral is now a rectangle.
Once again, we'll try drawing a rectangle inscribed inside a circle.
No matter how we draw our rectangle, this works!
Therefore (2) is sufficient and we can mark (B) as our answer (even if we don't understand why!)

** Note - (2) is true because we can set the diagonals of the rectangle as the diameter of the circle and the meeting point of the diagonals as its center.
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Re: Can a circle be drawn so that its circumference includes all four vert   [#permalink] 21 Dec 2017, 10:31
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