Question

find the value of z.

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.
Let the triangle be \(\triangle KNL\) with angle bisector \(KM\). According to the Angle - Bisector Theorem, \(\frac{z}{24}=\frac{14}{30}\)

Step2: Solve for \(z\)

Cross - multiply the proportion \(\frac{z}{24}=\frac{14}{30}\) to get \(30z = 24\times14\).
First, calculate \(24\times14=336\). Then, \(z=\frac{336}{30}\).
Simplify \(\frac{336}{30}\) by dividing both the numerator and the denominator by \(6\). So \(z=\frac{56}{5}=11.2\)

Answer:

\(z = 11.2\)