Two medians NA and OB of ΔNOP intersect each other at S at right angles. If NA = 15 cm and OB = 15 cm, then what is the length of OA?

$5\sqrt{5}$ cm

We know that,

The medians of a triangle intersect each other at the centroid, which divides each median in the ratio = 2:1.

Given,

Median NA = 15 cm

Median OB = 15 cm

As S is the point where the medians intersect, it divides each median in a 2:1 ratio, with the longer section towards the midpoint of the side. Therefore, the length of OS (longer section of OB) is 2/3 × 15 = 10 cm, and SA (shorter section of NA) is 1/3 × 15 = 5 cm.

Since the medians intersect at right angles, by the Pythagorean theorem, we can find OA as follows:

= OA² = OS² + SA²

= OA² = (10 cm)² + (5 cm)²

= OA² = 100 cm² + 25 cm² = 125 cm²

= OA = \(\sqrt {125}\) cm²

= OA = 5\(\sqrt {5}\) cm.