Question

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

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, $\angle ADC=\angle ABC={90}^{\circ }$, AC is the diameter of the circle.

$\Rightarrow$ AC $=2\times \text{radius}=2\times 4=8cm$