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 ___.

Answer 1

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 ${180}^{\circ }$.

i.e., $\angle ABC+\angle ADC={180}^{\circ }$.

But in parallelogram opposite internal angles are equal.

i.e., $\angle ABC=\angle ADC$

Using the above relations, we get $\angle ABC=\angle ADC={90}^{\circ }$.

A parallelogram with an internal angle of ${90}^{\circ }$ is called rectangle.

Since $\angle ADC=\angle ABC={90}^{\circ }$, ADC and ABC are semicircles, as angle in a semi-circle is ${90}^{\circ }$.

$\therefore$AC is the diameter of the circle.

$\therefore$AC = 2(radius) = 8 cm