, , are the medians of ΔABC. If BE = 12, CF = 9 and AB² + BC² + AC² = 600, BC = 10, find AD.

From the given info,

As AB² + BC² + AC² = 600

and BC = 10,

⇒ AB² + AC² = 500 ...... (1)

As AD is median,

⇒ BD = 5 ...... (2)

In Δ ABC, using the theorem of Apolloneous, we know that,

For AD to be the median, we have,

AB² + AC² = 2(AD² + BD²)

Substituting values for (1) and (2) in above equation,

⇒ 500 = 2 (AD² + 5²)

⇒ 250 = AD² + 25

⇒ AD² = 225

⇒ AD² = 15²

⇒ AD = 15