Question

n is the incenter of \\( \triangle rst \\). if \\( mn = 9x - 1, nl = 16x - 15 \\), find kn.

Explanation:

Step1: Use the property of incenter

The in - center \(N\) of a triangle is equidistant from the sides of the triangle. So \(MN = NL\).

$$9x−1=16x−15$$

Step2: Solve the equation for \(x\)

Subtract \(9x\) from both sides:

$$ - 1=16x - 9x-15$$

$$ - 1 = 7x-15$$

Add \(15\) to both sides:

$$14 = 7x$$

Divide both sides by \(7\):

$$x = 2$$

Step3: Find \(MN\) (and then \(KN\) since \(KN = MN\) as the in - center is equidistant from the sides)

Substitute \(x = 2\) into \(MN=9x - 1\)

$$MN=9\times2-1=18 - 1=17$$

Since \(KN = MN\) (property of in - center: distances from in - center to the sides are equal), \(KN = 17\)

Answer:

\(x = 2\)
\(KN=17\)