Question

in the diagram below, \\(\overline{gh}\\) is parallel to \\(\overline{de}\\). if \\(fh = 4\\), \\(fe = 7\\), and \\(gf = 4.8\\), find the length of \\(\overline{df}\\). figures are not necessarily drawn to scale.

Explanation:

Step1: Identify Similar Triangles

Since \( \overline{GH} \parallel \overline{DE} \), triangles \( FGH \) and \( FDE \) are similar by the Basic Proportionality Theorem (Thales' theorem). Thus, the ratios of corresponding sides are equal: \( \frac{FG}{FD} = \frac{FH}{FE} \).

Step2: Substitute Known Values

We know \( FG = 4.8 \), \( FH = 4 \), and \( FE = 7 \). Let \( FD = x \). Substituting into the proportion: \( \frac{4.8}{x} = \frac{4}{7} \).

Step3: Solve for \( x \)

Cross - multiply: \( 4x = 4.8\times7 \). Calculate \( 4.8\times7 = 33.6 \). Then, \( x=\frac{33.6}{4}=8.4 \).

Answer:

The length of \( \overline{DF} \) is \( 8.4 \).