Question

Intr-un triunghi dreptunghic ABC , unghiul A =90 grade , AD perpendicular BC, D apartine (BC) , se dau :
a) AB =16 cm si BC =20cm.Se cer BD,DC,AC,AD si Aria ABC
b)BC =50cm su BD =32cm.Se cer DC,AD,AB,AC si Aria ABC
c) AC =20cm si DC =16cm.Se cer BD,BC,AB,AD si Aria ABC​

Answer 1

Răspuns:

Explicație pas cu pas:

Intr-un triunghi dreptunghic ABC , unghiul A =90 grade , AD perpendicular BC, D apartine (BC) , se dau :

a) AB =16 cm si BC =20cm.Se cer BD,DC,AC,AD si Aria ABC

T. Pitagora AC=√(BC²-AB²)=√(20²-16²)=√(400-256)=√144=12   AC=12 cm.

T. catetei: AC²=BC·CD  ⇒144=20·CD  ⇒CD=144:20  ⇒CD=7,2 cm.

                AB²=BC·BD  ⇒256=20·BD  ⇒BD=256:20  ⇒BD=12,8 cm.

T. inaltimii: AD²=CD·BD=7,2·12,8=92,16  ⇒AD=√92,16=9,6 cm.

Aria ABC =cateta1·cateta2/2=12·16/2=6·16=96 cm²  Aria=96 cm²  

b) BC =50cm su BD =32cm.Se cer DC,AD,AB,AC si Aria ABC

DC=BC-BD=50-32=18 cm.  DC=18 cm.

T. inaltimii: AD²=CD·BD=18·32=576  ⇒AD=√576=24 cm. AD=24 cm.

AB²=AD²+BD²=24²+32²=576+1024=1600  ⇒AB=√1600=40  AB=40 cm.

AC²=BC²-AB²=50²-40²=2500-1600=900  ⇒AC=√900=30   AC=30 cm.

Aria ABC =cateta1·cateta2/2=30·40/2=600 cm²    Aria ABC=600 cm²

c) AC =20cm si DC =16cm.Se cer BD,BC,AB,AD si Aria ABC​

AD²=AC²-CD²=20²-16²=400-256=144  ⇒AD=√144=12   AD=12 cm.

T. inaltimii: AD²=CD·BD=16·BD  ⇒BD=12²/16=144/16=9  BD=9 cm.

BC=BD+DC=9+16=25   BC=25 cm.

T. Pitagora  AB=√(BC²-AC²)=√(25²-20²)=√(625-400)=√225=15  ⇒  

AB=15 cm.

Aria ABC =cateta1·cateta2/2=20·15/2=150 cm²  Aria ABC = 150 cm²