The chords AB and CD of a circle with center O are perpendicular to one diameter and intersect with this diameter

The chords AB and CD of a circle with center O are perpendicular to one diameter and intersect with this diameter at points M and N, respectively, what is the length of the segment MN, if it is known that | ОМ | = 4cm., | ОN | = 5cm.?

In this case, two options are possible:

The first option is when both chords lie on the same side of the center of the circle;

The second option is when the chords lie on opposite sides of the center of the circle.

Let’s consider the first option. In this case, the distance from the center of the circle to the points of intersection with the diameter must be subtracted. Let’s calculate this value:

5 – 4 = 1 cm.

Let’s consider the second option. In this case, the distances from the center of the circle to the intersection points must be added. Let’s count:

5 + 4 = 9 cm.

Answer: the length of the segment MN can be equal to 1 centimeter or 9 centimeters.