Find the value of x that makes N the incenter of the triangl

Find the value of x that makes N the incenter of the triangle.

Solution

If N is the incenter, then NL=KN=NM is the the Inradius of the circle.

In triangle NLC which is right angled at L, Then by pythagoras Theorem NC² = NL² + LC²

Therefore NL² = NC² - LC² = 52² - 48² = 2704 - 2304 = 400

Therefore NL = SQRT(400) = 20

Now NL = 20 is equal to NM = 4x, i.e, 4x = 20

Therefore x = 5