Question

in the diagram below, \\( \overline { k l } \\) is parallel to \\( \overline { h i } \\). if \\( h i = 7 \\), \\( j l = 3.6 \\), and \\( i l = 2.4 \\), find the length of \\( \overline { k l } \\). figures are not necessarily drawn to scale.

Explanation:

Step1: Use the similarity of triangles

Since \( \overline{KL}\parallel\overline{HI}\), \(\triangle JKL\sim\triangle JHI\) (by AA similarity, as \(\angle JKL=\angle JHI\) and \(\angle JLK = \angle JIH = 90^{\circ}\)).
For similar triangles, the ratio of corresponding sides is equal. That is \(\frac{KL}{HI}=\frac{JL}{JL + IL}\).

Step2: Substitute the given values

We know that \(HI = 7\), \(JL=3.6\), and \(IL = 2.4\).
Substitute these values into the proportion \(\frac{KL}{7}=\frac{3.6}{3.6 + 2.4}\).
First, calculate the denominator \(3.6+2.4=6\).
So the equation becomes \(\frac{KL}{7}=\frac{3.6}{6}\).

Step3: Solve for \(KL\)

Cross - multiply: \(6\times KL=3.6\times7\).
\(6KL = 25.2\).
Divide both sides by 6: \(KL=\frac{25.2}{6}\).
\(KL = 4.2\).

Answer:

\(4.2\)