Question

the figure below shows a section of a farm whose owner plans to divide it with line segment \\(\overline{fs}\\), which must be parallel to \\(\overline{mn}\\). \\(\overline{mn} = 18\\) m and \\(\overline{fs} = 12\\) m. given that \\(\overline{mt} = 42\\) m, how long is \\(\overline{mf}\\)?

Explanation:

Step1: Use the similarity of triangles

Since \( \overline{FS}\parallel\overline{MN}\), \(\triangle TFS\sim\triangle TMN\) (by the AA similarity criterion, as corresponding angles are equal). For similar triangles, the ratio of corresponding sides is equal. Let \(MF = x\), then \(FT=MT - MF=42 - x\). The ratio of the sides of similar triangles gives \(\frac{FS}{MN}=\frac{FT}{MT}\).

Step2: Substitute the known values

We know that \(FS = 12\), \(MN = 18\), and \(MT = 42\). Substituting into \(\frac{FS}{MN}=\frac{FT}{MT}\), we get \(\frac{12}{18}=\frac{42 - x}{42}\).
Cross - multiply: \(12\times42=18\times(42 - x)\).
First, calculate \(12\times42 = 504\) and \(18\times(42 - x)=756-18x\). So, \(504 = 756-18x\).
Add \(18x\) to both sides: \(18x+504 = 756\).
Subtract \(504\) from both sides: \(18x=756 - 504\).
\(18x = 252\).
Divide both sides by \(18\): \(x=\frac{252}{18}=14\).

Answer:

\(14\) m