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Question

ABCD is a parallelogram. If the radius of the circle passing through all the vertices of a parallelogram is 4cm, then the length of the diagonal AC is ___.

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Solution

Given, a circle is passing through all the vertices of the parallelogram ABCD.

Therefore ABCD is a cyclic parallelogram.

In a cyclic quadrilateral, sum of opposite internal angles is 180.

i.e., ABC+ADC=180.

But in parallelogram opposite internal angles are equal.

i.e., ABC=ADC

Using the above relations, we get ABC=ADC=90.

A parallelogram with an internal angle of 90 is called rectangle.

Since ADC=ABC=90, ADC and ABC are semicircles, as angle in a semi-circle is 90.

AC is the diameter of the circle.

AC = 2(radius) = 8 cm


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