Question

in △bcd, \\(\overline{bc} \parallel \overline{ef}\\). given that \\(db = 35\\), \\(de = 15\\), and \\(dc = 42\\), find \\(df\\).

Explanation:

Step1: Apply the Basic Proportionality Theorem (Thales' theorem)

Since \( \overline{BC} \parallel \overline{EF} \) in \( \triangle BCD \), by the Basic Proportionality Theorem, we have the proportion:
\( \frac{DE}{DB} = \frac{DF}{DC} \)

Step2: Substitute the given values

We know \( DE = 15 \), \( DB = 35 \), and \( DC = 42 \). Substituting these into the proportion:
\( \frac{15}{35} = \frac{DF}{42} \)

Step3: Solve for \( DF \)

Cross - multiply to get:
\( 35 \times DF = 15 \times 42 \)
First, calculate \( 15\times42 = 630 \). Then,
\( DF=\frac{630}{35} \)
Simplify \( \frac{630}{35} \), we divide 630 by 35. Since \( 35\times18 = 630 \), we get \( DF = 18 \).

Answer:

18