Question

this figure shows △lmn. nz is the angle bisector of ∠lnm. what is lz? enter your answer, as a fraction, in the box

Explanation:

Step1: Apply angle - bisector theorem

By the angle - bisector theorem in \(\triangle LMN\) with angle bisector \(NZ\) of \(\angle LNM\), we have \(\frac{LZ}{ZM}=\frac{LN}{MN}\).

Step2: Identify side - lengths

We know that \(LN = 4\), \(MN=3\), and \(LM=LZ + ZM\) where \(ZM = 5\). Let \(LZ=x\), then \(\frac{x}{5}=\frac{4}{3}\).

Step3: Solve for \(LZ\)

Cross - multiply to get \(3x = 20\), so \(x=\frac{20}{3}\).

Answer:

\(\frac{20}{3}\)