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Question

ABCD is a quadrilateral such that ABC+ADC =180 inside the quadrilateral.

Statement  1: the circumcircle of ΔABC intersects diagonal BD at D.

Statement 2: the circumcircle of ΔABC intersects BD at Dinside the quadrilateral.

Statement 3: the circumcircle of ΔABC intersects BD at D outside the quadrilateral.

Statement 4: the circumcircle of ΔABC does not intersect BD at all.

Statement 5: ABCD is called cyclic quadrilateral.


  1. Only statement 4 is true 

  2. Only statement 1 is true

  3. One of the statement 2 or statement 3 can be true

  4. Statement 1 and statement 5 are true


Solution

The correct option is D

Statement 1 and statement 5 are true



Let us assume the center of the circle is O. suppose the circle intersects BD at D.

We know that the angle subtended by a chord at the center is twice the angle subtended by it at any point on the circle. Now take the line segment AC which is clearly a chord of the circle.

As discussed above x = 2ABC, y = 2ADC.

But x and y form a complete angle so x+y = 360.

So we get ABC + ADC = 180 but given ABC + ADC = 180 which can only be satisfied if D and D coincide.

Thus the circumcircle intersects BD at D itself.

As A, B, C, D lie on the circle ABCD is called a cyclic quadrilateral.

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