Question

in \\( \triangle a b c, \overline{a b} \\| \overline{d e} \\). given that \\( c a=39, c d=21 \\), and \\( c b=52 \\), find \\( c e \\).

Explanation:

Step1: Use the basic proportionality theorem (Thales' theorem)

Since \(\overline{AB}\parallel\overline{DE}\) in \(\triangle ABC\), we have \(\frac{CD}{CA}=\frac{CE}{CB}\).

Step2: Substitute the given values

We know \(CA = 39\), \(CD = 21\), and \(CB = 52\). Substituting into \(\frac{CD}{CA}=\frac{CE}{CB}\), we get \(\frac{21}{39}=\frac{CE}{52}\).

Step3: Solve for \(CE\)

Cross - multiply: \(39\times CE=21\times52\). Then \(CE=\frac{21\times52}{39}\).
Simplify: \(21\times52 = 1092\), and \(1092\div39 = 28\).

Answer:

\(CE = 28\)