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总结帝笔记—初三寒假班第三讲
53
记住这个公式你就可以口算!
\text{海伦公式: } S_{\Delta} = \sqrt{P(P-a)(P-b)(P-c)}
(P为: P = \frac{a+b+c}{2} , a, b, c为三角形三边长)
\text{那么: } S_{\Delta DQC} = \sqrt{\frac{11}{2} \cdot \left(\frac{11}{2}-2\right) \cdot \left(\frac{11}{2}-4\right) \cdot \left(\frac{11}{2}-5\right)}
\left(p = \frac{2+4+5}{2} = \frac{11}{2}\right) \\ \begin{aligned} &= \sqrt{\frac{11}{2} \cdot \frac{7}{2} \cdot \frac{3}{2} \cdot \frac{1}{2}} \\ &= \frac{\sqrt{231}}{4} \end{aligned}
\begin{aligned} \text{又: } S_{\Delta DQC} &= \frac{1}{2} \times CQ \times DP \\ &= \frac{1}{2} \times 2 \times DP = DP \end{aligned}
\therefore DP = \frac{\sqrt{231}}{4} \text{ 根据 } \textcircled{1}.
S_{\text{四边形}ABCD} = 9 \times \text{高} = 9DP = \frac{9\sqrt{231}}{4}
\text{根据 } (\Delta) \frac{S_{\Delta MAD}}{S_{\Delta MAD} + \frac{9\sqrt{231}}{4}} = \frac{16}{25}
\Rightarrow S_{\Delta MAD} = 4\sqrt{231} \text{ (end!)}
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