The radius of the circumcircle of Δ PQR, if PN = 4 cm, QN = 5 cm, RN = 6 cm and ON = 2 cm, where O is the orthocentre is:

Concept:

The nine-point circle of a triangle is a circle going through 9 special points:

1) Midpoints of three sides of the triangle.

2) Three foot of the perpendicular from vertices to front side.

3) Three midpoints from the vertices to the orthocenter of the triangle.

Calculation:

Given, PN = 4 cm, QN = 5 cm, RN = 6 cm and ON = 2 cm

NP, NQ, NR are the distance of the centre of the nine-point circle to the vertices of the triangle

R is the circumradius of the triangle 

NO is distance of orthocentre to the centre of the nine point circle

According to the property of the nine-point circle:

NP² + NQ² + NR² + NO² = 3R²

⇒ 4² + 5² + 6² + 2² = 3R²

⇒ R² = 81/3 = 27

⇒ R = 3√3 cm