Question

in the diagram below, $\angle nml \cong \angle npo$. $pm = 8$, $no = 28$, and $np = 16$. find the length of $\overline{ol}$. round your answer to the nearest tenth if necessary.

Explanation:

Step1: Identify Similar Triangles

Since $\angle NML \cong \angle NPO$ and $\angle N$ is common to both $\triangle NML$ and $\triangle NPO$, by the AA (Angle - Angle) similarity criterion, $\triangle NML \sim \triangle NPO$.

Step2: Set Up Proportion for Similar Triangles

For similar triangles, the ratios of corresponding sides are equal. So, $\frac{NP}{NM}=\frac{NO}{NL}$. First, find $NM$. We know that $NP = 16$ and $PM=8$, so $NM=NP + PM=16 + 8=24$. Let $OL=x$, then $NL=NO + OL=28 + x$.

Substituting the known values into the proportion: $\frac{16}{24}=\frac{28}{28 + x}$.

Step3: Solve the Proportion

Cross - multiply: $16\times(28 + x)=24\times28$.

Expand the left - hand side: $16\times28+16x = 24\times28$.

Calculate $16\times28 = 448$ and $24\times28=672$. So, $448+16x = 672$.

Subtract 448 from both sides: $16x=672 - 448$.

$16x = 224$.

Divide both sides by 16: $x=\frac{224}{16}=14$.

Answer:

The length of $\overline{OL}$ is $14$.