a, Xét tam giác ABC có:
\(AC^2+AB^2=24^2+18^2=900=30^2=BC^2\)\(\Rightarrow\) Tam giác ABC vuông tại A
Xét tam giác ABC và MDC có:
\(\widehat{DMC}=\widehat{BAC}\)
\(\widehat{C}\) là góc chung
\(\Rightarrow\) Tam giác ABC ~MDC ( g.g)
b, Vì tam giác ABC~MDC \(\Rightarrow\dfrac{AB}{AC}=\dfrac{MD}{MC}=\dfrac{3}{4}\Rightarrow MD=\dfrac{3MC}{4}\)\(\dfrac{AC}{BC}=\dfrac{MC}{DC}=\dfrac{4}{5}\Rightarrow DC=\dfrac{5MC}{4}\)
Mà:
\(\dfrac{AB}{MD}=\dfrac{BC}{DC}=\dfrac{AC}{MC}=\dfrac{AB+BC+AC}{MD+DC+MC}=\dfrac{72}{\dfrac{3MC}{4}+\dfrac{5MC}{4}+\dfrac{4MC}{4}}\)\(=\dfrac{72}{\dfrac{12MC}{3}}\Rightarrow12MC=72.3=216\Rightarrow MC=18cm\)\(\Rightarrow MD=\dfrac{3.18}{4}=13,5cm\)
\(\Rightarrow DC=\dfrac{5.18}{4}=22,5cm\)
