Question

in the diagram, \\( \triangle a b c \sim \triangle d b e \\)
what is the length of \\( \overline{a b} \\) ?
the length of \\( \overline{a b} \\) is \\( \square \\) units.

Explanation:

Step1: Use the property of similar triangles

Since \(\triangle ABC\sim\triangle DBE\), the ratios of corresponding sides are equal. That is \(\frac{AB}{BE}=\frac{AC}{DE}\). Let \(AB = x\), then \(BE=x - 12\).

Step2: Substitute the values into the proportion

We know \(AC = 9\), \(DE = 5\). So \(\frac{x}{x - 12}=\frac{9}{5}\).
Cross - multiply: \(5x=9(x - 12)\).

Step3: Expand and solve the equation

Expand \(5x=9x-108\).
Subtract \(5x\) from both sides: \(0 = 9x-5x - 108\), so \(4x=108\).
Divide both sides by \(4\): \(x = 27\).

Answer:

\(27\)