The value of x for the given triangle is found to be 6.
Pythagoras theorem states that in a right angled triangle, the sum of the squares of the perpendicular and base is equal to the square of the hypotenuse.
It can be written as h² = p² + b².
Given triangle has N as the incentre.
Which implies that NJ = NL= NK.
Now, apply Pythagoras theorem in ΔANK to obtain,
AN² = AK²+ NK²
=> 37² = 35² + NK²
=> NK²= 37²- 35²
=> NK²= 144
=> NK = 12
Then, As per the question it can be written as,
2x = 12
=> x = 6
Hence, the value of x for the point N to be the incentre is 6.