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For how many of the following types of quadrilaterals does there exist

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For how many of the following types of quadrilaterals does there exist  [#permalink]

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New post 14 May 2019, 00:04
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Question Stats:

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For how many of the following types of quadrilaterals does there exist a point in the plane of the quadrilateral that is equidistant from all four vertices of the quadrilateral?

a square
a rectangle that is not a square
a rhombus that is not a square
a parallelogram that is not a rectangle or a rhombus
an isosceles trapezoid that is not a parallelogram


A. 1
B. 2
C. 3
D. 4
E. 5

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Re: For how many of the following types of quadrilaterals does there exist  [#permalink]

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New post 24 May 2019, 06:52
If there exist a point in the plane of the quadrilateral that is equidistant from all four vertices of the quadrilateral, then it must be cyclic.
Square, a rectangle that is not a square, and an isosceles trapezoid that is not a parallelogram are cyclic quadrilaterals.

Proof that Only parallelograms that are rectangle can be cyclic.
If ABCD is a parallelogram, then ∠A=∠C. If it is cyclic then [∠A+∠C=180]
Hence ∠A=90 and ∠C=90
Hence all cyclic parallelograms are rectangle.
Similarly we can prove that all cyclic rhombus are squares.
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Re: For how many of the following types of quadrilaterals does there exist  [#permalink]

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New post 25 May 2019, 02:54
Bunuel wrote:
For how many of the following types of quadrilaterals does there exist a point in the plane of the quadrilateral that is equidistant from all four vertices of the quadrilateral?

a square
a rectangle that is not a square
a rhombus that is not a square
a parallelogram that is not a rectangle or a rhombus
an isosceles trapezoid that is not a parallelogram


A. 1
B. 2
C. 3
D. 4
E. 5




Hi Could you please explain how you got 3 as the answer?
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Re: For how many of the following types of quadrilaterals does there exist  [#permalink]

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New post 30 May 2019, 17:49
Square, Rectangle - Center 2
Rhombus sym - diag diff, Paralleogram - diagonals different,
Isosceles trapezoid - diag same / symmetry of sides - 1

For the visually inclined, take a square and stretch out the bottom.
The point equidistant from each vertex follows the axis.
Easiest to visualize is three equilateral triangles together

C - 3
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Re: For how many of the following types of quadrilaterals does there exist   [#permalink] 30 May 2019, 17:49
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For how many of the following types of quadrilaterals does there exist

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