Question

in the diagram below, $overline{no}$ is parallel to $overline{kl}$. if $lo = 9$, $no = 16$, and $kl = 28$, find the length of $overline{mo}$. figures are not necessarily drawn to scale. state your answer in simplest radical form, if necessary.

Explanation:

Step1: Prove similar triangles

Since $\overline{NO}\parallel\overline{KL}$, $\triangle MNO\sim\triangle MKL$ (by AA - similarity, as $\angle M$ is common and $\angle NOM=\angle KLM = 90^{\circ}$ because of parallel lines and right - angled sides).

Step2: Set up proportion

The ratios of corresponding sides of similar triangles are equal. So, $\frac{MO}{ML}=\frac{NO}{KL}$. Let $MO = x$, then $ML=x + 9$. We have $\frac{x}{x + 9}=\frac{16}{28}$.

Step3: Cross - multiply

Cross - multiplying gives $28x=16(x + 9)$.

Step4: Expand and solve

Expand the right - hand side: $28x=16x+144$. Subtract $16x$ from both sides: $28x-16x=144$, so $12x = 144$. Divide both sides by 12: $x = 12$.

Answer:

$12$