Question

in the figure below, the line segment \\( \overline{rq} \\) represents one of the sides of a farm. a natural fence is required along that length. the lengths of \\( \overline{ps} \\), \\( \overline{pt} \\), and \\( \overline{tr} \\) are known and measure 24 yards, 15 yards, and 7.5 yards, respectively. what is the length of the side to be fenced with natural fencing, if \\( \overline{ps} \\) and \\( \overline{qr} \\) are parallel?

Explanation:

Step1: Use the similarity of triangles

Since \( \overline{PS}\parallel\overline{QR}\), \(\triangle PTS\sim\triangle QTR\).
The ratio of corresponding sides of similar triangles is equal. That is \(\frac{PT}{TR}=\frac{PS}{QR}\).
We know \(PT = 15\) yards, \(TR=7.5\) yards, and \(PS = 24\) yards.

Step2: Solve for \(QR\)

Substitute the known values into the proportion \(\frac{PT}{TR}=\frac{PS}{QR}\).
We get \(\frac{15}{7.5}=\frac{24}{QR}\).
Cross - multiply: \(15\times QR=7.5\times24\).
Then \(QR=\frac{7.5\times24}{15}\).
Calculate \(7.5\times24 = 180\), and \(\frac{180}{15}=12\).

Answer:

The length of the side to be fenced (\(QR\)) is \(12\) yards.