Question

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.
diagram of a right triangle with right angle at l, a segment no parallel to kl, and right angle at o for triangle mon
answer
\\(mo = \\)
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Explanation:

Step1: Identify similar triangles

Since $\overline{NO} \parallel \overline{KL}$, $\triangle MON \sim \triangle LIK$ by AA similarity (corresponding angles equal).

Step2: Set up proportion

Corresponding sides: $\frac{MO}{LI} = \frac{NO}{KL}$. Let $MO = x$, $LI = LO + OM = 9 + x$.

Step3: Solve for $x$

$\frac{x}{9 + x} = \frac{16}{28} = \frac{4}{7}$. Cross multiply: $7x = 4(9 + x) = 36 + 4x$.

Step4: Calculate $x$

$7x - 4x = 36 \implies 3x = 36 \implies x = 12$.

Answer:

12