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Math Expert V
Joined: 02 Sep 2009
Posts: 55801
For how many of the following types of quadrilaterals does there exist  [#permalink]

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Difficulty:   75% (hard)

Question Stats: 36% (01:40) correct 64% (01:24) wrong based on 47 sessions

HideShow timer Statistics 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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Director  P
Joined: 19 Oct 2018
Posts: 565
Location: India
Re: For how many of the following types of quadrilaterals does there exist  [#permalink]

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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.
Manager  B
Joined: 07 May 2018
Posts: 54
Re: For how many of the following types of quadrilaterals does there exist  [#permalink]

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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?
Intern  B
Joined: 23 Mar 2019
Posts: 3
Re: For how many of the following types of quadrilaterals does there exist  [#permalink]

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