Lời giải:

Áp dụng hệ thức lượng trong tam giác vuông:

$AB^2=BH.BC$

$AC^2=CH.CB$

$\Rightarrow \frac{9}{16}=\frac{BH}{CH}=(\frac{AB}{AC})^2$

$\Rightarrow \frac{AB}{AC}=\frac{3}{4}$

$AC=\frac{4}{3}AB=\frac{4}{3}.24=32$ (cm)

$BC=\sqrt{AB^2+AC^2}=\sqrt{24^2+32^2}=40$ (cm)

$AH=\frac{AB.AC}{BC}=\frac{24.32}{40}=19,2$ (cm)

Hình vẽ:

(AB/AC)^2=HB/HC

=>(AB/AC)^2=9/16

=>AB/AC=3/4

Bài 2: 

Ta có: \(\dfrac{HB}{HC}=\dfrac{1}{3}\)

nên HC=3HB

Ta có: \(AH^2=HB\cdot HC\)

\(\Leftrightarrow HB^2=48\)

\(\Leftrightarrow HB=4\sqrt{3}\left(cm\right)\)

\(\Leftrightarrow BC=4\cdot HB=16\sqrt{3}\left(cm\right)\)

Bài 1:

ta có: \(AB=\dfrac{1}{2}AC\)

\(\Leftrightarrow\dfrac{HB}{HC}=\dfrac{1}{4}\)

\(\Leftrightarrow HC=4HB\)

Ta có: \(AH^2=HB\cdot HC\)

\(\Leftrightarrow HB=1\left(cm\right)\)

\(\Leftrightarrow HC=4\left(cm\right)\)

hay BC=5(cm)

Xét ΔBAC vuông tại A có AH là đường cao ứng với cạnh huyền BC

nên \(\left\{{}\begin{matrix}AB^2=HB\cdot BC\\AC^2=HC\cdot BC\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}AB=\sqrt{5}\left(cm\right)\\AC=2\sqrt{5}\left(cm\right)\end{matrix}\right.\)

Bài 2: 

Xét ΔABC có 

\(BC^2=AB^2+AC^2\)

nên ΔABC vuông tại A

Xét ΔABC vuông tại A có AH là đường cao ứng với cạnh huyền BC, ta được:

\(\left\{{}\begin{matrix}AB^2=BH\cdot BC\\AC^2=CH\cdot BC\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}BH=\dfrac{25}{13}\left(cm\right)\\CH=\dfrac{144}{13}\left(cm\right)\end{matrix}\right.\)

Bài 1: 

Ta có: \(\dfrac{AB}{AC}=\dfrac{5}{6}\)

\(\Leftrightarrow HB=\dfrac{25}{36}HC\)

Ta có: \(AH^2=HB\cdot HC\)

\(\Leftrightarrow HC^2\cdot\dfrac{25}{36}=900\)

\(\Leftrightarrow HC=36\left(cm\right)\)

hay HB=25(cm)

\(1,\dfrac{AB}{AC}=\dfrac{5}{6}\Leftrightarrow AB=\dfrac{5}{6}AC\)

Áp dụng HTL tam giác

\(\dfrac{1}{AH^2}=\dfrac{1}{AB^2}+\dfrac{1}{AC^2}\Leftrightarrow\dfrac{1}{900}=\dfrac{1}{\dfrac{25}{36}AC^2}+\dfrac{1}{AC^2}\\ \Leftrightarrow\dfrac{1}{900}=\dfrac{36}{25AC^2}+\dfrac{1}{AC^2}\\ \Leftrightarrow\dfrac{1}{900}=\dfrac{36+25}{25AC^2}\Leftrightarrow\dfrac{1}{900}=\dfrac{61}{25AC^2}\\ \Leftrightarrow25AC^2=54900\Leftrightarrow AC^2=2196\Leftrightarrow AC=6\sqrt{61}\left(cm\right)\\ \Leftrightarrow AB=\dfrac{5}{6}\cdot6\sqrt{61}=5\sqrt{61}\\ \Leftrightarrow BC=\sqrt{AB^2+AC^2}=61\left(cm\right)\)

Áp dụng HTL tam giác: 

\(\left\{{}\begin{matrix}AB^2=BH\cdot BC\\AC^2=CH\cdot BC\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}BH=\dfrac{AB^2}{BC}=...\\CH=\dfrac{AC^2}{BC}=...\end{matrix}\right.\)

Bài 1:

Ta có: \(\dfrac{AB}{AC}=\dfrac{5}{6}\)

\(\Leftrightarrow HB=\dfrac{25}{36}HC\)

Ta có: \(AH^2=HB\cdot HC\)

\(\Leftrightarrow HC^2\cdot\dfrac{25}{36}=900\)

\(\Leftrightarrow HC=36\left(cm\right)\)

hay HB=25(cm)

Bài 2: 

Xét ΔABC có \(BC^2=AB^2+AC^2\)

nên ΔABC vuông tại A

Xét ΔABC vuông tại A có AH là đường cao ứng vói cạnh huyền BC, ta được:

\(\left\{{}\begin{matrix}AB^2=BH\cdot BC\\AC^2=CH\cdot BC\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}BH=\dfrac{25}{13}\left(cm\right)\\CH=\dfrac{144}{13}\left(cm\right)\end{matrix}\right.\)

AB/AC=4/3

=>HB/HC=16/9

=>HB/16=HC/9=k

=>HB=16k; HC=9k

AH^2=HB*HC

=>144k^2=24^2=576

=>k=2

=>HB=32cm; HC=18cm

AB=căn 32*50=40cm

AC=căn 18*50=30cm

\(1,HC=\dfrac{AH^2}{BH}=\dfrac{256}{9}\\ \Rightarrow AB=\sqrt{BH\cdot BC}=\sqrt{\left(\dfrac{256}{9}+9\right)9}=\sqrt{337}\\ 2,BC=\sqrt{AB^2+AC^2}=10\left(cm\right)\\ \Rightarrow BH=\dfrac{AB^2}{BC}=6,4\left(cm\right)\\ 3,AC=\sqrt{BC^2-AB^2}=9\\ \Rightarrow CH=\dfrac{AC^2}{BC}=5,4\\ 4,AC=\sqrt{BC\cdot CH}=\sqrt{9\left(6+9\right)}=3\sqrt{15}\\ 5,AC=\sqrt{BC^2-AB^2}=4\sqrt{7}\left(cm\right)\\ \Rightarrow AH=\dfrac{AB\cdot AC}{BC}=3\sqrt{7}\left(cm\right)\\ 6,AC=\sqrt{BC\cdot CH}=\sqrt{12\left(12+8\right)}=4\sqrt{15}\left(cm\right)\)

Ta có: \(\dfrac{AB}{AC}=\dfrac{5}{7}\)

nên \(\dfrac{HB}{HC}=\dfrac{25}{49}\)

hay \(HB=\dfrac{25}{49}HC\)

Ta có: \(AH^2=HB\cdot HC\)

\(\Leftrightarrow HC^2=15^2:\dfrac{25}{49}=441\)

\(\Leftrightarrow HC=21\left(cm\right)\)

\(\Leftrightarrow HB=\dfrac{75}{7}\left(cm\right)\)

Hệ thức lượng trong tam giác vuông :

\(AB^2=BC.BH\left(1\right)\)

\(AC^2=BC.CH\left(2\right)\)

\(\left(1\right):\left(2\right)\Rightarrow\dfrac{BH}{CH}=\dfrac{AB^2}{AC^2}=\dfrac{25}{36}\left(\dfrac{AB}{AC}=\dfrac{5}{6}\right)\)

\(\Rightarrow BH=\dfrac{25}{36}CH\)

mà \(AH^2=BH.CH\)

\(\Rightarrow\dfrac{25}{36}CH^2=AH^2=30^2\)

\(\Rightarrow\dfrac{5}{6}CH=30\Rightarrow CH=\dfrac{30.6}{5}=36\) (\(\left(cm\right)\)

\(\Rightarrow BH=\dfrac{25}{36}.36=25\) \(\left(cm\right)\)

Xét tg vuông ABH và tg vuông ACH có

\(\widehat{BAH}=\widehat{ACH}\) (cùng phụ với \(\widehat{ABC}\) )

=> tg ABH đồng dạng với tg ACH

\(\Rightarrow\dfrac{AH}{HC}=\dfrac{HB}{AH}=\dfrac{AB}{AC}=\dfrac{5}{6}\)

\(\Rightarrow\dfrac{30}{HC}=\dfrac{5}{6}\Rightarrow HC=\dfrac{6.30}{5}=36cm\)

\(\Rightarrow\dfrac{HB}{30}=\dfrac{5}{6}\Rightarrow HB=\dfrac{5.30}{6}=25cm\)