a) AD ĐL Pi-ta-go vào \(\Delta ABC\) vuông tại A:
\(\Rightarrow BC^2=AB^2+AC^2\)
\(=5^2+12^2=169\)
\(\Rightarrow BC=\sqrt{169}=13\left(cm\right)\)
AD CT: \(AB.AC=AH.BC\)
\(\Rightarrow5.12=AH.13\Rightarrow AH\approx4,62\left(cm\right)\)
\(sin\widehat{B}=\frac{AC}{BC}=\frac{12}{13}\)
\(\Rightarrow\widehat{B}\approx67,22^o\)
\(\Rightarrow\widehat{C}\approx90^o-67,22^o=22,78\)
b) AD t/c đường p/g trong tam giác:
\(\Rightarrow\frac{BC}{BK}=\frac{AC}{AK}=\frac{BC+AC}{BK+AK}=\frac{13+12}{AB}=\frac{25}{5}=5\)
\(\Rightarrow\frac{13}{AK}=5\Rightarrow AK=2,6\left(cm\right)\)
c) Kẻ \(DF//AB\left(F\in AC\right)\Rightarrow DF\perp AC\)
\(\Rightarrow\Delta ADF\) vuông cân tại F
\(\Rightarrow DF=FA\Rightarrow AD^2=DF^2+FA^2=2DF^2\)
\(\Rightarrow AD=\sqrt{2}.DF\Rightarrow DF=\frac{AD}{\sqrt{2}}\)
Vì \(DF//AB\Rightarrow\frac{DF}{AB}=\frac{FC}{AC}=\frac{AC-FA}{AC}\)
\(\Rightarrow\frac{DF}{AB}=1-\frac{FA}{AC}\Rightarrow\frac{DF}{AB}+\frac{FA}{AC}=1\Rightarrow\frac{DF}{AB}+\frac{DF}{AC}=1\)
\(\Rightarrow\frac{1}{AB}+\frac{1}{AC}=\frac{1}{DF}\Rightarrow\frac{1}{AB}+\frac{1}{AC}=\frac{\sqrt{2}}{AD}\)