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A quadrilateral ABCD is inscribed in a circle. Are the diagonals AC

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A quadrilateral ABCD is inscribed in a circle. Are the diagonals AC  [#permalink]

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New post 14 Jul 2018, 11:11
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43% (00:55) correct 57% (01:03) wrong based on 20 sessions

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A quadrilateral ABCD is inscribed in a circle. Are the diagonals AC and BD equal?

(1) ABCD is a parallelogram.

(2) AB = BC.
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A quadrilateral ABCD is inscribed in a circle. Are the diagonals AC  [#permalink]

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New post 14 Jul 2018, 11:56
amanvermagmat wrote:
A quadrilateral ABCD is inscribed in a circle. Are the diagonals AC and BD equal?

(1) ABCD is a parallelogram.

(2) AB = BC.


Statement-1:-
A cyclic quadrilateral has its opposite angles supplementary(sums to 180 degree).
A parallelogram has its opposite angles equal, hence the angles of the parallelogram are 90 degree each. So, the parallelogram under discussion is a rectangle.
And the diagonals of rectangle are equal in length.(Since they are the diameters of the circle & their intersection point is the center of the circle)

Sufficient.

Statement-2:- AB=BC
So, Diagonals would be equal when pair of two opposite sides are equal(With given AB=BC; if AB=CD, BC=AD or AB=BC=CD=AD). That means the possible case is when the quadrilateral is a square or rhombus. Again we can deduce that an inscribed rhombus is a square in the same line of reasoning we did for st1.

In all other cases, diagonals are not equal. (With AB=BC, locate D at multiple positions on the arc AC, we will get numerous quadrilaterals with different diagonals)

Insufficient.

Ans. (A)
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A quadrilateral ABCD is inscribed in a circle. Are the diagonals AC &nbs [#permalink] 14 Jul 2018, 11:56
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