[tex]\it Fie\ \Delta ABC-isoscel,\ AB=AC, \ h_a= 20,\ h_b=h_c=24.
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Fie\ AD \perp BC,\ D\in\ BC,\ \ \Rightarrow cos C = \dfrac{\dfrac{a}{2}}{b}\ \ \ (1)
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h_a\cdot a=h_b\cdot b \Rightarrow 20a = 24b \Rightarrow \dfrac{a}{b} =\dfrac{\ 24^{(4}}{20} \Rightarrow \dfrac{a}{b} = \dfrac{6}{5}|_{:2} \Rightarrow
\dfrac{\dfrac{a}{2}}{b} = \dfrac{3}{5}\ \ \ (2)
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(1),( 2) \Rightarrow cos C = \dfrac{3}{5}[/tex]
[tex]\it sin C=\sqrt{1-cos^2C} =\sqrt{1-\dfrac{9}{25}} =\sqrt{\dfrac{16}{25}} =\dfrac{4}{5}\ \ \ \ (3)
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Din\ \Delta DCA \Rightarrow sin C=\dfrac{AD}{AC} \Rightarrow sin C = \dfrac{h_a}{b} \Rightarrow sin C = \dfrac{20}{b} \ \ \ (4)
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(3),\ (4) \Rightarrow \dfrac{20}{b} = \dfrac{4}{5} \Rightarrow b=\dfrac{20\cdot5}{4} \Rightarrow b = 25
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h_a\cdot a=h_b\cdot b \Rightarrow 20\cdot a = 24 \cdot25 \Rightarrow 20\cdot a= 600|_{:20} \Rightarrow a=30[/tex]
