Question

find the value of m.

Explanation:

Step1: Apply the Angle - Bisector Theorem

The Angle - Bisector Theorem states that if a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the adjacent sides. In \(\triangle LKN\) with angle bisector \(KM\), we have \(\frac{LM}{MN}=\frac{LK}{NK}\).

Step2: Substitute the known values

We know that \(LM = m\), \(MN = 8\), \(LK = 25\), and \(NK = 20\). Substituting these values into the proportion \(\frac{m}{8}=\frac{25}{20}\).

Step3: Solve for \(m\)

Cross - multiply: \(20m=25\times8\).
First, calculate \(25\times8 = 200\). So, \(20m = 200\).
Then, divide both sides by 20: \(m=\frac{200}{20}\).

Answer:

\(m = 10\)