Question

find the length of \\(\overline{ab}\\).
\\(\overline{ab} = \square\\)

Explanation:

Step1: Use the mid - segment theorem

Since \(EB\) is a mid - segment of \(\triangle ADC\) (because \(E\) and \(B\) are mid - points as \(AE = 12, ED=12\) and \(BC = 18, AB\) is to be found), by the mid - segment theorem of a triangle (\(EB=\frac{1}{2}DC\)), we can also use the proportion \(\frac{AE}{AD}=\frac{AB}{AC}\). Here \(AD=AE + ED=12 + 12 = 24\), \(AC=AB + BC\). Let \(AB=x\), then \(AC=x + 18\).

Step2: Set up the proportion

\(\frac{AE}{AD}=\frac{AB}{AC}\), substituting \(AE = 12\), \(AD = 24\), \(AB=x\) and \(AC=x + 18\) gives \(\frac{12}{24}=\frac{x}{x + 18}\).
Cross - multiply: \(12(x + 18)=24x\).
Expand: \(12x+216 = 24x\).
Subtract \(12x\) from both sides: \(216=24x - 12x\).
Simplify: \(12x = 216\).
Divide both sides by \(12\): \(x=\frac{216}{12}=18\).

Answer:

\(18\)