初中数学八年级上册试题库（555题）

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如图，在\(\Delta ABC\)中，点\(D\)在射线\(BC\)上，过\(AC\)的中点\(E\)作线段\(FG\)交\(AB\)于点\(G\)，连接\(CF\)，且\(\angle B = \angle DCF\)，\(CF = 6\)，\(AC = BC = 10\)，\(AG = 3BG\)，求\(\Delta ABC\)的周长。

知识点：第十二章　全等三角形

参考答案：证明：\(\because \angle B = \angle DCF\)，\(\therefore CF//AB\)，\(\therefore \angle FCA = \angle A\)，\(\angle F = \angle FGA\)，\(\because \)点\(E\)是\(AC\)的中点，\(\therefore AE = EC\)，在\(\Delta AEG\)和\(\Delta CEF\)中，\(\left\{ {\begin{array}{*{20}{l}}
{\angle A = \angle FCA} \\
{\angle FGA = \angle F} \\
{AE = EC}
\end{array}} \right.\)，\(\therefore \Delta AEG \cong \Delta CEF(AAS)\)。\(\because \Delta AEG \cong \Delta CEF\)；\(\therefore CF = AG = 6\)，\(\because AG = 3BG\)，\(\therefore BG = 2\)，\(\therefore AB = 8\)，\(\therefore \Delta ABC\)的周长\( = AB + AC + BC = 28\)。