Question

In triunghiul ABC, <A =90 de grade si AD perpendicular pe BC, D apartine lui BC. Daca AD= 6 cm si DC= 4cm, calculati BD, AB si AC.
VA ROG AJUTĂ-MĂ!!! ​

Answer 1

Explicație pas cu pas:

teorema înălțimii:

AD² = BD•DC

6² = BD•4 => BD = 9 cm

BC = BD + DC = 9 + 4 = 13 => BC = 13 cm

teorema catetei:

AB² = BD•BC = 9•13 => AB = 3√13 cm

AC² = DC•BC = 4•13 => AC = 2√13 cm

Answer 2

$\it \Delta ADC-dreptunghic,\ \widehat{D}=90^o,\ Th.\ Pitagora\ \Rightarrow AC^2=AD^2+DC^2=\\ \\ =6^2+4^2=36+16\ =4(9+4)\ =4\cdot13\ \Rightarrow AC=\sqrt{4\cdot13}=2\sqrt{13}\ cm$

$\it Th.\ catetei\ \Rightarrow AC^2=BC\cdot DC\ \Rightarrow (2\sqrt{13})^2=BC\cdot 4\ \Rightarrow 4\cdot13=BC\cdot4\Big|_{:4}\ \Rightarrow \\ \\ \ \Rightarrow BC=13\ cm\\ \\ BD=BC-DC=13-4=9\ cm\\ \\ Th.\ Pitagora\ pentru\ \Delta DAB\ \Rightarrow AB^2=AD^2+BD^2=6^2+9^2=36+81=\\ \\ =117\ \Rightarrow AB=\sqrt{117}=\sqrt{9\cdot13}=3\sqrt{13}\ cm$